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AD-AI06 740 AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OH F/B 20/11
FINITE ELEMENT MODEL OF A WHITE-METZNER VISCOELASTIC POLYMER ETC(U)
FEB 81 B R COLLINSUNCLASSIFIED AFIT-CI-91-31T
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A FINITE ELEMZNT MODEL
Or A WHITE-METZNER VISCOELASTIC
POLYMER EXTRUDATE
by
BRENT R. COLLINS
Submitted to the Department of Aeronautics and Astro-
nautics on January 16, 1981 in partial fulfillment of the
requirements for the degree of Master of Science.
'4 ABSTRACT
A finite element model of a viscoelastic polymer melt
characterized by a White-Metzner rheological equation of
state was developed. For creeping flow wherein the inertia
terms are negligible non-linear finite element equations were
solved by a method of direct substitution termed Picard itera-
tion.
Four flow geometries were examined: cross channel, plane
couette, entry, and step flow. A comparison of two bi-quadradic
isoparametric element types (8 node "serendipity" and 9 node
1Lagrange') showed general superior behavior of the Lagrange
elements. The *penalty* method of incompressible flow was used
with the Galerkin method to formulate the finite element equation,
yielding satisfactory behavior for creeping inelastic and visco-
elastic flow.
The non-linear equations yielded numerical convergence up
to Weissenberg numbers of 0.01. Techniques of expanding this
radius of convergence were examined and proposed for future effort.
Thesis Supervisor: Dr. David K. Roylance
Title: Associate Professor of Materials Engineering /
i
81- 31T
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RESEARCH TITLE: A Finite Element Model of a White-Metzner Viscoelastic Polymer Extrudate
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FOLD IN
A FINITE ELEMENT MODEL
OF A WHITE-METZNER VISCOELASTIC
POLYMER EXTRUDATE
by
BRENT R. COLLINS
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 1981
o Brent R. Collins 1981The author hereby grants to M.I.T. permission to reproduce andto distribute copies of this thesis document in whole or in part.
Signature of Author duwt eo aeBrent R. Collins, Department of Aeronautics
and Astronautics
Approved by _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
oy &. Schluntz, oechnical Supervisor, CSDL
Certified by T44 .AuPDavid K. Roylande, Thesis Supervisor
I'! P ,I. .
Accepted byHarold Y. Wchman, Chairman, Departmental Graduate
Committee
A FINITE ELEMENT MODEL
OF A WHITE-METZNER VISCOELASTIC
POLYMER EXTRUDATE
by
BRENT R. COLLINS
Submitted to the Department of Aeronautics and Astro-
nautics on January 16, 1981 in partial fulfillment of the
requirements for the degree of Master of Science.
ABSTRACT
A finite element model of a viscoelastic polymer melt
characterized by a White-Metzner rheological equation of
state was developed. For creeping flow wherein the inertia
terms are negligible non-linear finite element equations were
solved by a method of direct substitution termed Picard itera-
tion.
Four flow geometries were examined: cross channel, plane
couette, entry, and step flow. A comparison of two bi-quadradic
isoparametric element types (8 node "serendipity" and 9 node
"Lagrange") showed general superior behavior of the Lagrange
elements. The "penalty" method of incompressible flow was used
with the Galerkin method to formulate the finite element equation,
yielding satisfactory behavior for creeping inelastic and visco-
elastic flow.
The non-linear equations yielded numerical convergence up
to Weissenberg numbers of 0.01. Techniques of expanding this
radius of convergence were examined and proposed for future effort.
Thesis Supervisor: Dr. David K. Roylance
Title: Associate Professor of Materials Engineering
lawi
TABLE OF CONTENTS
PAGE
Abstract . . . . ................. i
Table of Contents ...... ............... ii
List of Figures ................ ..... iii
Acknowledgements ..... ............... . iv
I. Introduction ...... ................. . 1
II. The Finite Element Method in Fluid Dynamics . 5
III. Viscoelastic Constitutive Models . ....... . 13
IV. Viscoelastic Finite Element Equations ..... ... 22
V. Computer Implementation .... ............ . 41
VI. Calculation Results ..... .............. . 46
Cross Channel Flow .... ............ . 46
Plane Couette Flow .... ............ . 49
Entry Flow ...... ................ . 51
Step Flow ...... ................. ... 55
VII. Model Evaluation ............... 57
VIII. Conclusions ....... .................. .. 61
IX. Recommendations ...... ................ . 63
APPENDIXES
Appendix 1: Derivation of Elastic Stress GradientExpression . . ........... 83
Appendix 2: Calculation of the Global SecondDerivatives ... ........... . 86
Appendix 3: Listing of New Subroutines . . .. 91
Appendix 4: Input Dataset Listings ...... 109
Appendix 5: Brief Review of Gyroscope Theory . 121
Bibliograpy .. .................... .125
ii
-
LIST OF FIGURES
Page
1. Piece Part Assembly of A Plastic Gyroscope. 65
2. Typical Injection Molding Process. 66
3. Fluid Maxwell Element. 67
4. Non-Uniform Molecule Mesh for Solving Stress 68Gradients.
5. Calculation of Elastic Stresses at Gauss Points. 69
6. Iteration Schemes for the Solution of Creeping, 70Viscoelastic Finite Element Equations.
7. FEAP Flowchart. 71
8. Flow Geometries and Boundary Conditions. 72
9. Velocity Flow Field for Linear Cross Channel Flow. 73
10. Velocity Comparisons of 9- and 8-Node Elements for 74Cross Channel Flow.
11. Computed Pressures in Cross Channel Flow Field. 75
12. Fully Developed Flow Behavior of Viscoelastic Fluid 76
Entering and Leaving a Contracting Channel.
13. Uniform Inlet Velocity Flow Behavior of Newtonian 77Fluid Entering and Leaving a Contracting Channel.
14. Fully Developed Flow Behavior of Newtonian Fluid 78Entering and Leaving a Contracting Channel.
15. Velocities and Pressures for Newtonian Entry Flow. 79
16. Elements of a Complete Injection Molding Flow Analysis 80
17. Solutions to Non-Linear Equations Accession 81
18. Cutaway of Gyroscope. NTIS GRA&I 82r DTIC TAB
TABLE Ulannounced 0Justification-
Computer Run Matrix 82a
Distrliution/IV
iii -i
.1.
ACKNOWLEDGEMENTS
I am most grateful to Professor David K. Roylance for
his timely guidance of my work.
I would also like to acknowledge the financial support
of the Charles Stark Draper Laboratory which gave me the
opportunity to pursue this project and provided funding for
the computer time under IR & D Contract No. 18339. I also
wish to acknowledge the sponsorship of the United States Air Force
under the direction of the Air Fcrce Institute of Technology.
I am pleased to acknowledge "Kid" Benny Fudge, Billy White,
and Mitch Hansberry who gave me no help at all.
iv
_______iv
I. INTRODUCTION
The finite element method has become a popular numerical
tool in the analysis of fluid flow problems (1, 2, 3, 4, 5).
Particularly in the regime of incompressible flow this method
has become very competitive with the more established finite
difference methods due to its simplicity of implementation
and generality of handling mixed boundary conditions for com-
plex geometries which favor nonuniform meshes at points of
singularities. Accordingly, the finite element discretization
process is used herein to characterize the flow of a polymeric
melt under various geometric conditions. The particular approach
is the Galerkin weighted residual formation of the non-symmetric
integral equations (6,2), with the penalty method used for the
pressure term via an approximation of the incompressible con-
tinuity equation (1,3,7). Steady, two dimensional flow is treated
and viscoelastic fluid effects are modeled by employing an
Oldroyd codeformational stress derivative in a modified Maxwell
constitutive equation (8).
The motivation for this analysis stems from the development
of low cost, medium performance, plastic gyroscopic instruments
at the Charles Stark Draper Laboratory. With the exception of
the momentum wheel and electromagnetic parts, a complete single
degree of freedom integrating gyroscope has been designed using
glass filled polyphenylene sulfide parts. Performance goals
are in the range of 1 - 10 degrees/hour drift rate. Cost ad-
vantages derive from elimination of precision secondary machining
I
of metallic components as well as basic material costs. However,
the need for uniform physical and mechanical properties (e.g.,
dimensional stability, thermal conductivity, mechanical compli-
ance) of the parts to provide the required performance after
possible long-term storage in unair-conditioned warehouses
necessitates the correlation of the final material state to
processing parameters. Such knowledge will permit the rationale
selection of extrusion parameters, post fabrication treatments,
and subsequent analysis of storage and service environmental
effects on instrument performance. Figure 1 shows a picture
of the typical plastic gyroscope under consideration. Figure 2
depicts a typical injection molding process for these gyroscopic
parts.
Roylance (9) has pointed out that the information the
engineer is seeking in a flow analysis is the location of regions
of elevated shear deformation, which can lead to mechanical de-
gradation and higher residual stresses, regions of stagnation
and recirculation, at which overlong material residence and
thermal degradation might occur, and power requirements for the
fabrication process itself. Also of interest for the gyroscope
application are the effects of the flow field on the distribution
of the filler fibers which are carried along by the drag of the
fluid. It is possible that the zones of filler depletion or
enhancement which are observed in molded parts, can be predicted
and controlled by evaluation of the calculated velocity field.
2
- - - 0-
.r
In the above regard the numerical analysis of polymer
melts cai be broken down into two general categories. First is
the evaluation of the accuracy of the solution themselves.
Calculations are made and compared to known exact or approximate
analytic solutions. Typical of these are the pipe/channel flow,
drag flow, and die entry flow. By far most of the numerical
studies have been in this category. In the second category is
the application of numerical solutions to real problems. Only
three studies are known to this author which have aimed at apply-
ing numerical results to actual polymer processing. The first
is the work of Bigg (10) who used the Marker and Cell Finite
Difference Scheme to specify preferred operations for the mixing
of polymer solutions in a single screw extruder. The second is
the National Science Foundation/Industry supported work at Cornell
University (10), also using finite difference methods to evaluate
mold filling and control the location and orientation of weld
lines. Thirdly is the work of Caswell and Tanner (12) who effec-
tively used the finite element method to redesign wire coating
dies to eliminate recirculation.
The current work falls into the first category described
above, but the intention of applying the numerical model, once
assessed for accuracy and utility, is kept firmly in mind and
discussed throughout this report. To conclude the introduction,
it is also necessary to describe how the current analysis fits
into the completely general solution. In the injection molding
process, the flow is non-steady and non-isothermal (but approxi-
mately adiabatic within the fluid boundaries), with advancing
free surfaces until the mold is completely filled. Upstream of
3
the flow front the fluid is completely surrounded by either
rigid boundaries or adjacent fluid. For an incompressible
fluid, a complete numerical model must therefore account for
unsteady, non-isothermal, free surface effects. In addition,
the observance of a finite recoverable shear in the rheological
data of polymer melts indicates the nee to include viscoelastic
effects in the model. For unsteady effects, since the Reynolds
number (Re) of flow is always much less than unity, a good ap-
proximation is achieved by ignoring inertia and employing the
linear "creeping" flow solution. The model that we are eventually
striving for then is an adiabatic, viscoelastic solution with
changing surface boundaries. Time is included only as temperature
is conducted and convected and as the velocity field is perturbed
by the changing boundary. The current work investigates the
viscoelastic effects with the simplifying assumptions of two
dimensional, steady state flow.
To this end, this report contains a brief review of the
finite element method, a discussion of the viscoelastic consti-
tutive models used in the finite element equations, the details
of the numerical schemes used in solving the equations, the
computer implementation of the numerical schemes, a discussion
of calculations conducted for four flow geometries to assess
the numerical model, and an evaluation of the application of
the numerical technique to the gyroscope fabrication.
4
- . . .. --•-- . ... ...."'" - .. ... . " ... .,, j , . .. .. _ '. ,, . . .L . ; .. . ...... . ... . .J,,/L :_ _ ]. -tl . -,
II. THE FINITE ELEMENT METHOD IN FLUID DYNAMICS
This section is not intended to be exhaustive in nature,
but rather to review some of the more important features of
the finite element method employed in this work. References
may be consulted for a more thorough treatment of the methods.
We begin by repeating that the finite element method is
an approximate method of solving the differential equations of
boundary and initial value problems (1,2). Field variables are
solved by dividing the bounded region into subsets (finite
elements) which themselves are governed by the differential
equations. By approximating the distribution of the variables
within each finite element by a trial function, the variables
at any point in the element can be determined by a linear com-
bination of the variable at specified points on the element
edges. These points are called the nodes of the element; the
variables at the nodes being determined by solving linear alge-
braic equations formed by assembling all of the elements into
a matrix equation of order pqm, where p is the number of elements,
q is the number of nodes per element, and m is the number of
variables per node. The coefficients of the variables in the
simultaneous equations are the integrals of the governing dif-
ferential equation taken over the region of the element which is
bounded by that node.
Mathematically, we write the discretization as:
Fdn - E FdYi 0 (11.i)
Yi
5
with prescribed boundary conditions. In equation II.1, F is
the governing differential equation, Q is the entire region
and 7i is the region of the finite element. Where physical
relations apply (such as the virtual work principle in solid
mechanics), the equations can be formed in that basis. This
is the approach used in references 1 and 9.
When the differential equation is self-adjoint (can be
written in the form (py')' + qy + f = 0) with appropriate
boundary conditions the equations can be formed by an abbreviated
variational principle by merely multiplying the differential
equation by the variation of the independent variables, i.e.
f (py') + qy + f)6ydYi = SI = 0 (11.2)
Yi
where I is the integral of the variational problem formed from
the governing differential equation. Of course, this is merely
stating that the euler equation of the variational principle
is identical to the governing differential equation (see [13)).
When the equations are not self adjoint, or the boundary condi-
tions are unsuitable, an extremum principle can still be found,
unless odd number derivatives are present. In that case, which
is the situation with the complete Navier-Stokes equation with
convection, a true extremum principle does not exist [14].
Pormation of the finite element equations by a variational
principle is the Ritz method. This method is most useful for
the "creeping" flow solution of viscous fluids where the govern-
ing differential equation is known to be the euler equation of
the proper extremum principle (15].
6
In the case of the complete Navier-Stokes equation, the
method of weighted residuals is used wherein the error which
remains after substituting appropriate trial functions into
the governing equation is orthogonally projected to a set of
weighting functions [2]. By setting the inner product of the
error and the weighting function equal to zero, the approximate
differential equation is then satisfied. Zienkiewicz (1] de-
scribes the two most popular methods of selecting the weighting
functions as the Galerkin and Collocation methods. Due to its
generality, the Galerkin method is the most popular for formu-
lating the finite element equations for fluid flow problems.
Selecting this method then the element variables are approximated
by m
a = Z N.C. (11.3)j=l J 3
where a is the field variable in the element, C. are the values
of the variable at the node points and N. are the set of trial
(basis) functions which satisfy the element boundary condition.
When equation 11.3 is substituted into the functional F of equation
II.1, we obtain in general:
n i niZl F(a)dyi E f edyi 0 (II.4)
7
. .5 .
where c is the residual error of the differential equation.
Now using the Galerkin method of forming the inner product
of the error and the trial functions we obtain:
nmNkF(E NC)dYi - 0 (k-l,m) (11.5)
Yi
In this manner, we form m times n equations for the determi-
nation of the value of variable a at the points CJ.
In selecting the field variables to be approximated
Frecaut [16) provides an excellent review of the advantages and
disadvantages of the different formulations. The governing
equations in an eulerian reference frame are continuity
V-u 0 (11.6)
and momentum
p[u,t + (uVu b0 - Vp + V.0 (II.7)
where in rectilinear flow:
u is the velocity vector
bb is the body force vector by
-- bz]
[ji0
0 is the deviatoric stress vector Lo2li + 022)i + a23-k
! 31 + 0 32) + a 3k
p is the constant density
p is the hydrostatic pressure
V is the gradient operator -i + +ay_+ ! 39 -'
and the comma denotes differentiation with respect to time.
If the flow is purely viscous, the deviatoric stresses can be
written as explicit functions of the velocity gradients leaving
only velocity and pressure as independent variables. If both
are approximated by the Galerkin method, the number of unknowns
is relatively high (i.e. components of velocity at each node
plus the pressure). In addition, some of the diagonal terms
of the coefficient matrix become zero which limits the pivoting
techniques generally used for solving the equations.
Two methods have been devised for eliminating the pressure.
For two dimensional flow, the stream function u = *,y and v = - ,x
is used to satisfy continuity and results in the disappearance
of the pressure term when inserted into the momentum equation.
However, the application to mixed boundary value problems is
difficult, as shown by Tanner [17]. For incompressible problems,
the penalty function formulation has been developed. This method,
reviewed in detail in [7]. replaces the incompressible continuity
equation by the approximation
p = -a (V-u) (I1.8)
9
-~~~~~~~"os in ---- -------p - -
where a is a large positive number whose effect is to "penalize"
the error of not satisfying continuity. In reference [7], it is
shown that this method converges to the exact solution for "creep-
ing" flow and that the selection of a is determined from the
relation:
a = cu (11.9)
Where c is a constant equal to 107 and ji is the dynamic viscosity.
Furthermore, to avoid the trivial solution of u - 0 as a -
(see equation II.10) the coefficients determined from evaluation
of the integral must be singular. This is accomplished by employ-
ing reduced integration (quadrature rule of lower order than
the exact for a given element) for the pressure term. The other
terms can then be integrated at the optimum order (selective
reduced integration) or at the lower order (uniform reduced in-
tegration). While it is more accurate to employ selective
reduced integration (SRI), it is usually more convenient to use
uniform reduced integration (URI) in the computer programs.
Since it has been shown that 8 node quadrilateral elements
exhibit inferior behavior to 9 node elements even for SRI, it
is strongly recommended that when URI is used the 9 node
"Lagrange" isoparametric element be employed [7].
Bercovier (18] has recently concluded that the reduced
integration approach is only valid for straight-sided elements
(biquadratic) if the governing equation is linear ("creeping"
flow) and valid only for rectangular elements (vice bilinear
quadrilaterals) when the equation is non-linear (with convection).
Since most of our work concerns linear systems, this is not
viewed as a limitation.
10
For ease of implementation, economy, and accuracy, therefore,
we selected the penalty method with URI, 9 node Lagrange isopara-
metric elements. For comparison, some eight node "serendipity"
element cases were run and will be discussed in Section VI.
Applying the Galerkin formulation of the finite element
equations we obtain the following for two dimensional, recti-
linear, incompressible, viscous flow:(A + +_K) _+ M u + f = o (11.10)
Where
u is a column vector of the two dimensional velocities
at the node points,
N is the matrix of trial (shape) functions,
KD DB dyf= R = -d
~= fp N" (V*(N u)T)TN dyE " j (r fm) Ta mT B dy
• NTZd
and
f -j f T bo dy r VT t dr
11I
- .-- ----.---.--------- .--...
In the matrix definitions above, we used the further identi-
ties:
r20 01,D= 0 , 2 0 g = , where1001 J
L is the differential operator matrix for two dimensional
flow aax 0
Lay ax
Also the second term in the expression for f is the surface
traction on the line element r which results from integrating
the viscous stress term by parts. (Throughout this report a
single underline denotes a vector quantity, and a double under-
line denotes a matrix quantity.)
When the inertial effects are comparable to the viscous
ones, i.e., Reynolds No. greater than one, equations II.10 are
non linear and must be solved by some iterative scheme. A
discussion of these techniques will be postponed until the non-
linear viscoelastic effects are added in Section IV.
Of course equation II.10 is the well known "weak" form
of the Navier-Stokes governing differential equation which has
been derived elsewhere by the virtual work statement Il].
12
. .. .. .. . . .+ . . .- ,- . - - + ,,.-'
III. VISCOELASTIC CONSTITUTIVE MODELS
The selection of a viscoelastic constitutive model (the
rheological equation of state) for use in the finite element
equations is generally a compromise between the accuracy of the
model and ease of implementation. Because all of the models are
nonlinear consideration must be given to the relative effects
on the numerical convergence of the solutions. In this study,
two general ground rules were used in selecting the appropriate
model. First, for the material under consideration (fiber-filled
polyphenylene sulfide), adequate rheological or viscometric data
do not exist to justify the use of multiple constant models, and
second only a first order effect on the flow field was being
sought. Once success is achieved in modeling viscoelasticity,
rheological data can be obtained and adjustments to the consti-
tutive model investigated.
As before, only essential elements for understanding the
behavior of the selected viscoelastic model are presented in
this report. For a thorough discussion of the continuum mechanics
of viscoelastic materials the references can be consulted (19,
20, 21, 22, 23).
For a fluid element, the resistance to deformation when a
force is applied can be thought of as a combination of viscous
and elastic stresses. Modeling these as a dashpot and spring
respectively as shown in Figure 3, we obtain the well-known
Maxwell Element for fluids. Using the nomenclature of Figure 3,
13
where u is the dynamic viscosity, G is the shear modulus of
elasticity, e is the infinitesimal strain and a is the applied
shear stress we obtain the stress-strain rate relation:
£ G +A (111.1)
Generalizing to a three-dimensional form, we have:
a+ A Tt ()= 2d (111.2)
where a is the Cauchy deviatoric stress tensor
U is the dynamic viscosity
X = U/G is a time constant known as the relaxation time
and d is the rate of deformation tensor whose components are
defined as:
dij - ' + au (i,j = 1,2,3) (111.3)
Equation 111.2 is suitable when the rate of deformation
in the flow is infinitesimually small. But for general motion,
in which the rate of deformation is not necessarily small, the
time derivative of the Cauchy stress tensor violates two funda-
mental requirements of any equation of state. These requirements
are that the equation describes material properties independent
of the frame of reference, and that the behavior of any material
element must depend only on its previous deformative history and
not in any way on the state of ne jhboring elements, or on rigid
body translation/rotation. These discrepancies are corrected by
substituting for the time derivative of the Cauchy stress either
an Oldroyd derivative [8] (known as a convected or a codeforma-
tional derivative) or a Jaumann derivative [24] (known as a
14
co-rotational derivative). These modifications will be
discussed shortly. Once the above requirements are satisfied
it only remains to tailor the equation so as to fit experimental
observations. This is done by introducing added parameters
which are multiplied by functions of the invariants of the rate
of strain tensor.
Han (23] presents a survey of the major refinements developed
for the two invariant stress derivatives along with the material
properties they predict. A two constant (A,V) model using an
Oldroyd derivative is known as a White-Metzner model. When the
Jaumann derivative is used, the equation is called a DeWitt model.
As multiple parameters are added, the general models are known
merely as n-order Oldroyd models. Two other models derived by
means somewhat different from the generalized Maxwell element
are the Spriggs model which builds many Maxwell elements at the
molecular structure level and the Rivlin Erickson fluid which
merely states that the fluid stress is a function of the invariants
of the gradients of displacement, velocity, acceleration, second
acceleration, and so on.
Returning to the invariant stress derivatives, we write them
explicitly for further discussion. For the Oldroyd derivative
in contravariant form (see (22] for a discussion of covariant
and contravariant tensors) we obtain:
u" aiJ ack all. ak xkJ u.Wo.= i + t - a . ax k (111.4)
t y kk axk xk
Where the range and summation indicial convention is used.
15
.. -------.- - -I.., .. . _-.
Similarly the Jaumann derivative is:
Pa.. aau..+ uk 1 + Wik ak A + wjk ik (111.5)
Vt at kaxk
1 aui au.where 2ij ax_- u. are the components of the vorticity
tensor. Again, see Han [23] for an excellent discussion of the
physical significance of the terms on the right-hand side of
equations III.4 and 111.5.
We will also have occasion to discuss further the Rivlin-
Ericksen fluid so we list the general equation for an incom-
pressible fluid:
a A + 2 1+a3A +a4 A 2 +a( A +A)(,1 i(1) a2 A(1) +3 A(2) 4 (2) +a5 (A(1 )A(2 ) + (22 A 2 A 2
(A( 1 )A( 2 ) + A(2)A I))+7 (A( 2 )A( 1 ) + A(l)(2)
2 A2 2 2 2+oL8 (A ( )A (2) + A( 2 ) + A( 2 )A( 1 )) (111.6)
where the a. are functions of the invariants of A(1) and A(2)
and(1)
Aij = 2dij
Aij(1) (1) auk !1) aukAij a l' + a + (i) +A(
In passing, it is noted that the preceding discussion of
models has focused on the rate type. If equation 111.2 is inte-
grated with respect to time rheological equations of state of
the integral type are obtained. While this type proves useful
for some rheological investigation, it complicates finite element
calculations by requiring a complete time history of the strain
path of all elements.
16
j; 6
Finally, before we can discuss the relative merits of the
models, we must make some definitions. Steady simple shear
flow, also known as viscometric flow, is defined by the velocity
field
u = ly, v = w = o (111.7)
where $ is a constant shear strain rate and
u is the velocity normal to the y axis of the cartesian
coordinate system.
Substituting equation 111.7 into 111.3 we find the rate of
deformation tensor to be:
[ 2 o 0 (111.8)
For viscometric flow, viscoelastic fluids are observed to
exhibit three independent material properties, the standard
viscosity, and a first and second normal stress function written
consecutively as:
T12 = ('Y,°all - a22 = 1 (y)y , a22 - 033 = 42 (yy
(111.9)
Implicit in equations 111.9 is the further observation that
when a fluid behaves viscoelastically, the material parameters
are not constant, but vary with the rate of strain. This non-
newtonian behavior is generally observed to follou a power law
relation, written for the viscosity as:
17
-1 (1-n) (III.10)1 + (K/po) (y/ 2)
where Uo, K, and n are parameters selected emperically. When
the exponent n is less than zero, the viscosity varies inversely
to the shear strain rate and the fluid is termed shear-thinning.
When n is greater than zero the fluid is shear thickening. Most
real fluids are shear thinning. t1 and *2 on the other hand are
observed to increase exponentially with shear strain rate.
Before we continue, recall that equation III.10 was written
for simple shear flow. This equation is merely the specializa-
tion of the more commonly written general flow form:
Uo1(Id) = -1 (l-n)/2 (III.11)
1 + (Klpo) d )
where IId is the second invariant of the rate of strain tensor
IId = dijdij, (111.12)
which in two dimensional rectilinear flow can be written explic-
itly as
rau 2 _ v 2] 2- ud + k + 2 IV+ a(1
We are now prepared to make a selection of the constitutive
equation to implement in the finite element equations. The
choices have been narrowed to (i) White-Metzner (ii) DeWitt and
(iii) Rivlin-Ericksen as generally representative of the avail-
able models (Pipkin and Tanner (25] present a thorough review
of all the models for viscosmetric flow). Middleman [26] has
18
presented an excellent discussion of the correlation of the
properties predicted by the White-Metzner and DeWitt models
to experimental observations. In simple shear flow, the DeWitt
model is somewhat superior because the second normal stress func-
tion is finite whereas the White-Metzner model predicts that it
vanishes. However, in general flow fields the DeWitt model
varies appreciably from reality while the White-Metzner model
maintains consistency. Since *2 is generally small, the fact
that the White-Metzner model predicts a zero value is not con-
sidered a major drawback by Middleman and we agree. Han [23]
suggests that since the Oldroyd derivative takes a different
form if written in terms of covariant or contravaraint basis
vectors that it is inferior to the Jaumann derivative. Since
the work herein is conducted for a rectilinear coordinate
system, it is felt that this is less of a penalty than the
cited deviation of the Jaumann derivative model for general flow
fields. Therefore, the author concurs with Middleman's recom-
mendation that the White-Metzner model is preferred to the
DeWitt model.
Considering the Rivlin-Ericksen fluid, Tanner [17) notes
that for a simple shear flow equation 111.6 reduces to:
a. *= oA (i) + OP' + (1) (1) - l Ai 2 ) (111.11)
1 1 + 4#2 "Aik ki 3
Clearly equation III.6 is overly complicated for our initial
work. But sin.a the simplification to 111.11 presumes simple
shear flow, it is disqualified as a candidate for this effort.
It is interesting to note, however, that of the three models
19
considered, the Rivlin Ericksen fluid alone permits the de-
viatoric stress to be written as an explicit function of the
velocities and nth order derivatives of velocities. The ad-
vantages of this fact will become obvious in the next section
when we discuss the formation and solution of the complete
finite element ecuation.
Let us recapitulate before concludina this section. A
White-Metzner modified Maxwell element was selected for the
rheological ecuation of state because of its ability to approxi-
mate real viscoelastic fluid behavior while requiring only two
model parameters. In addition, the two parameters V and X are
taken to be functions of the second invariant of the rate of
strain tensor as defined in equation III.11.
For plane, steady flow where w = W= g= o, the nine
equations of 111.2 reduce to four which are written explicitly
below with the use of equation III.4.x + + x + v 3u xx B a y x 2u U_ (III.12a)
,aca a au au av\_auavxy -ax By XXBy xy xy ay ax
(III.12b)
yy =y w - -2 - x w x -g 2
(III.12d)
20
These equations are identical to those used by Perera and Strauss
[27] in their finite difference formulation of similar problems
when account is made of the reduction of the four-constant model
they used vice the two parameter model used herein.
The reader is reminded that the stresses in equations 111.12
are the deviatoric ones and differ from the complete stresses by
the hydrostatic pressure. Since the momentum equation always
expresses these two stresses separately, they are not combined
here either.
21
& . . . . ". . .. .=- "- .. . .. - . . -
IV. VISCOELASTIC FINITE ELEMENT EQUATIONS
The governing differential equations for an incompressible
viscoelastic fluid are as presented in equations 11.6 and 11.7.
Continuity and momentum are repeated:
V'u - o (IV.la)
p [ut + (u - V)u] - b_o - Vp + V.a_ (IV.1b)
The boundary conditions of course will be for the independent
variables and gradients of these variables. However, for many
flow problems, it is more convenient to specify the tractions
(stresses) on some boundaries and the independent variables on
others. This is the mixed boundary condition formulation and
is of course mandatory for finite element equations which are
reduced to a set of inhomogeneous linear algebraic equations.
While specification of the variables (u, *, p, a depending on
the type of equations used) at the boundaries is straight
forward, the specification of boundary tractions must be con-
sistent with the type of problem. For example, Chang [15)
discusses the difference in specifying the surface traction,
for a number of flow cases, between a non-newtonian viscous
fluid and a generally viscoelastic one. Understanding these
differences is particularly important when a specific type of
flow is prescribed (e.g., fully developed entry flow) for an
assessment of the accuracy of the finite element model. We
defer further comment on the boundary conditions until Section VI
when specific flow problems are considered.
22
- - --- ----- -.-. ,
Briefly reviewing the past work on finite element modeling
of viscoelastic flow, it is noted that no investigations, known
to the author, have been conducted using the "penalty" method
for incompressible fluid flow. Tanner [17J and Caswell and
Tanner [12) have used the formulation with velocities and pressure
as the independent variables, with a Rivlin-Ericksen fluid for
viscometric flow. Results have been excellent for power law
fluids, but only Poiseuille flow has been considered for the
viscoelastic case. Kawahara and Takeuchi [28] applied a mixed
method where the total deviatoric stress (viscous and elastic)
was independently interpolated along with the velocities and
pressure. The White-Metzner constitutive equation was then
solved simultaneously with the Navier-Stokes equation for in-
compressible fluids. Using six-node triangular elements in
plane flow, this gives rise to eighteen additional unknowns
per element and is felt to have limitations for general problems
because of the computer capacity required for large, complicated
geometric problems. However, they did achieve good results for
expanding and bending flow through channels for relaxation times
up to 0.1 seconds.
In the work most similar to the current effort, Chang et. al.
[15] solved the equations using the White-Metzner model with
velocities and pressure the field variables for the finite ele-
ment equations. In two-dimensional, steady state, convective,
isothermal flow, the slip stick problem was solved for material
cases of Weissenberg numbers up to 0.2.
23
The Weissenberg number is a dimensionless ratio of re-
coverable or elastic shear stress to total shear stress in
steady flow. It is written
WS = XU (IV. 1)
L
where X is the relaxation time in seconds,
U is a characteristic steady velocity in cm/sec,
and L is a characteristic length in cm.
Han [23] presents rheological data for high and low density
polyethylene at various shear strain rates (U/L) at 200°C.
For high density polyethylene, the Ws varies from 35 at 0.025
cm/cm-sec down to 0.01 at 100 cm/cm-sec. On the other hand,
the Ws for low density polyethylene varies between 5 at the low
strain rate and 0.01 at the high strain rate. We note that this
is essentially the range of interest for practical problems
(0.01<Ws<35). A major difficulty in the finite element method
has been obtaining numerical convergence for problems of rela-
tively, high Ws as evidenced in the above review. It appears
that Chang's work has provided the highest value. Without dis-
cussion, it is noted that with this convergence problem, the
added numerical problems associated with evaluation of the
pressure term in the tangent stiffness matrix for the penalty
method may suggest some limitations in the future for applica-
tion to viscoelasticity.
Now using the Galerkin formulation with the penalty method,
equations IV.l become for steady state
p(V*(N u)T)TN + (mT B)T B mT B)dv L 1dv-o (IV.2)
v v
24
- - - - *. - - ---- - S
Where all terms have been defined in equation II.10, the
body forces are assumed to be zero, and two-dimensional recti-
linear flow is treated so that the plane stress vector a is:
S yy (IV. 3)
.ax y
We now split the deviatoric stess into a viscous and elastic
portion
a= av + ae (IV.4)
substitute into .equation IV.2, and apply Green's divergence
theorem to obtain
~ (~B+NP(V.u) T)TN + (mTB )TamTt)v1~
v v (IV.5)
where the viscous stress has been written explicity as
a=DLN U DBu (IV.6)
and the last term is the traction on the boundary. From equation
111.2 we can writeha(Ov + AOe)+ I = 2 (IV.7)
or since av = 2uij
ae = - A (IV.8)
where e is the 2D rate of deformation vector.
25
- - - --- ~ .--- - ------------- 7
From equations 111.12, we see that for steady state equation
IV.8 is of the following functional form
e = g(u, ae , av , 1 , , U', X, y) (IV.9)
Where the prime denotes differentiation with respect to x and y.
But since av is a unique function of u' we can further state
e = h Ue eu. = h(u, u', u x, y, a , a ). (IV.10)
Equation IV.10 now makes equation IV.5 not only non-linear
(even for creeping flow), but inexpressible in an explicit form.
The equation must, therefore, be solved simultaneously with
equation IV.5. This is the same point reached by Chang [15]
and Perera [27]. Let us examine the method of solution proposed
in [15]. Although convection was included in that analysis, it
is easier to consider creeping flow (without loss of generality).
The creeping, viscoelastic flow can be written as:^ e^ A ^ e qe'
K u + K (u, u , , , )=f (IV.ll)
where the terms Ke are the functional form of the internal elastic
forces. Newton-Raphson iteration can not be employed to solve
IV.l1 because of the implicit dependent variable ae. Instead
the common method is to use successive substitution where an
initial value of ae is guessed and substituted into equation
IV.10. Assuming u has first been solved for the linear problem,
Ke can now be calculated, substituted into equation IV.l1 andA P
a new value of u found. This new value of u is than used with
the latest value of ae to calculate an updated value of ge and
the process is repeated until some convergence criterion is
26
satisfied. In terms of a solution for u at iteration s+l, we
have:
K us+l + Ke
es = (is ^S -I'S eS-i e' S-1)
and as, u , x, e (IV.12)
The actual calculation on the computer was performed at the iteration
es S+ls+l by subtracting E from f and solving K u s . Therefore,
the computer equation is:
A f Ke s K ^
where Au = S- u. If the convection non-linearity is in-
cluded the Picard substitution can be nested within a Newton-
Raphson iteration.
If we momentarily disregard the issue of convergence, the
only problem which remains is the calculation of the elastic
stress gradient at the s-1 iteration. Chang [15] is completely
silent on this issue and it is felt that it was ignored. Later
on, we will discuss possible situations where this might be valid.
To aid in the discussion, let us write equation 111.12 in vector
form by recognizing axy = a yx It can be verified that the equa-
tion becomes:
e = A A - (u V)a] (IV.13)
r
27
L-T7
129u 0 2 au
0
Where A 0 23V 2a
av Du (au +av
and aa Do(u • V)a = u + v
For one-dimensional flow equation IV.13 becomes
aaae = e _ u -- xx+b(I.4xx xx x.
auwhere a and b are the appropriate functions of u and It is
convenient to use this equation to discuss the methods of solution
for the first order non-linear differential equation.
Equation IV.14 is the identical form of the Picard methode
of first order euuations namely [29]: dx F(x,y) where a cor-
responds to y and a, u, and b are functions only of x. The
equation is integrated yieldingx
Y = YO +f F(x,y)dx
xo
where yo is the initial value at x0
Equation IV.14 would become:
x
ae ae 1 r(a-1)a + b dx (IV.15)xx XXo 0 u I
28
[_ .1,
Assuming the integral could be evaluated numerically
ae could be solved by the same successive substitution schemexx
used for the complete finite element equations. An initial
guess is made for ae in the integrand and the right-hand sidexx
is solved for an updated value of axxe That value is then
substituted into the integrand and the procedure repeated until
convergence is achieved. Let us now write IV.13 in this form
1 e(u.V)a x (a -A a), (IV.16)
and upon integrati6n by taking the dot product of both sides
with dA = dxi + dy_A
(u + v)af = a( + (e - A ).dA (IV.17)
Ab
While in theory, IV.17 could be solved, it is felt that in a finite
element formulation, it would be impractical to use such a system
that requires an initial value to be calculated at a corner of
each element (ao) and separate integration of the spatial derivatives,
i.e.,
A x y Af 1 - A ) =f ae A E)dx +f QeA f -A2a)dxdy
A0 x0 YO Ao
Due to the difficulties encountered, another method was
sought for the solution of IV.13. If the derivative is approxi-
mated by a Taylor series, then a standard finite difference
equation is achieved and usual relaxatica methods can be employed
for the solution. Referring to Figure 4 and using central dif-
ferences we have for the first component of ae
29
-swum&
ij = - .- 1 ij .u ij. -e 2xx Wx xx
j 2 2- u j 11i 3 u ' , i i,j ad2i,j u i ' 3 xx v i' j 3a xaxy xay
(IV.18)
Where
and
(IV. 19)
Equations IV.19 are derived in Appendix I.
A few words about equations IV.18 and IV.19 are in order.
While central differences are expected to give higher order
accuracy, Roache [30] notes that the numerical stability is much
poorer than backward (upwind) differencing and for a non-uniform
mesh (special mesh system), it is very likely that the approxi-
mation deteriorates from third order accuracy in the mesh point
spacing to first order accuracy. In IV.18, the viscous stress
is expressed in terms of the equivalent rate of strain through
30
. ...- - - - - - - ---- _____
equation IV.6. Also the expressions for the gradient of viscous
stresses appear to treat the dynamic viscosity as independent of
x and y. This is not the case. Rather it can be seen upon
Sjudifferentiation of the products a (xx,y) for example, that
for a power law fluid the term (h ) is of higher order and,
therefore, is neglected. Finally all terms in IV.19 are elastic
xx stresses. The subscripts and superscripts have been dropped
so as to not severely encumber the equations. Equation IV.18
is a first order derivative counterpart to the steady, convection-
dissipation finite difference equation which gives rise to
classic under and over-relaxation methods. However, we do not
have an equivalent Courant number so we merely employ Richardson/
Jacobi iteration. Calling the left-hand side of IV.18 iteration
k+l and the elastic stress terms on the right-hand side iteration
k (which is known) we sweep through the entire solution domain
ein the relaxation process. As in most cases, a at the first
iteration k=l is assumed to be zero. The issues which we must
discuss in solving IV.18 by this technique are the selection of
mesh points i,j, evaluation of the second derivative of the
velocity, convergence of the iteration, and treatment of boundary
elements where boundary conditions must be invoked. We will take
these in the listed order.
Since all the terms involving the field variable u in
equation IV.18 are routinely calculated, in the evaluation of
the integrals of the finite element equations, at the Gauss
points in the Gauss quadrature it is natural to select these
points as the mesh for the elastic stresses. Then for the
31
differences required in evaluating the elastic stress gradients,
the elastic stress at Gauss points of adjacent elements can be
used. This procedure is shown in Figure 4 for one of the Gauss
points. Of course, two concerns arise. Procedurally, most
finite element routines calculate element quantities such as
velocity gradients in subroutines which dump the values upon
exiting the subroutine, returning only values of global tangent
stiffness components. Therefore, special schemes must be devised
to identify, maintain, and pass current values of elastic stresses,
external to the subject element, to the element for an update of
the elastic stress at its Gauss points. Second, the discontinuity
of stresses between ;elements which gives rise to the practice of
"smoothing" must be recognized. At the early stages of itera-
tion, this might aggravate the numerical stability. For this
study, the first issue was resolved by programming techniques
(principly by creating arrays which were stored in common
memories between subroutines). The second issue was not ad-
dressed.
For the problem of the evaluation of the second derivative
(recall from Section II we are using a "weak" form of the equa-
tions so that only CO continuity is required of u), we now
require C1 continuity of the trial functions and explicitly
evaluate the term just as is done for the first derivative.
To do this, a subroutine was written (ESHAP) which returnsah vle -x2 2N i , 9?Ni 3
the values ,2 y y at the Gauss points of an element.
32
The values -, etc. are then calculated in the exact same3x
2
manner as done for the first derivatives. For this subroutine
of course, it was also necessary to calculate the determinant
of the Jacobian of the second derivatives. The mathematics
involved in subroutine ESHAP are given in Appendix 2.
Considering for the time being only convergence of the
Richardson/Jacobi iteration scheme, (Newton-Raphson dnd Picard
iteration are briefly treated later). We can apply the Lax/
Richtmyer amplification matrix error method [31] discussed in
[2]. Briefly, we write equation IV.13 in terms of the final
value and errors at each iteration or
(a +e )k+l v( + e + E)-(uV)(av + a e + J
(IV. 20)
Subtracting IV.13 from IV.20 we get
_k+l = - k (IV.21)
or
Ek+l- = - u.V)<l (IV.22)
The test for convergence then is for the eigen values of the
matrix X(A - u-V) to be <1. Note that the dimensions of this
tridiagonal matrix are 3np where n is the number of Gauss points
per element and p is the number of elements. The complete
matrix is formed by assembling the individual 3x3 matrices
at each Gauss point. We did not conduct any further analysis
33
of convergence, but rather have established bounds emperically.
Little emphasis was placed on this issue because it was found
during the course of the study that the outer iteration of
equation IV.12 generally controlled convergence.
Finally at the boundary elements where an adjacent
element may not exist, it is necessary to devise an auxiliary3,e aae
scheme for the calculation of - and g-- at the Gauss points.
If Oe is known at the element edges (in particular the node
points) the nodes can be used as the forward or backward mesh
points and the relaxation procedure continued. However, there
are some major drawbacks to this. First regardless of the
boundary condition (velocity or traction specified) additional
calculations for velocity gradients and viscous= stress gradients
at the nodes must be accomplished. Additionally, the elastic
stress gradient can not employ central differences at the node,
but must be based on a backward difference. Third, the forma-
tion of the two independent equations to simultaneously solve9,e aae
and - is quite cumbersome. A different technique wasTX- ay
therefore developed.
A new common array was established (BOSIG) to identify
and pass the elastic stresses at the four corner nodes. At
the first iteration, these stresses (four nodes by three stress
components by the number of elements) are initialized at zero.
The velocity vector u is then calculated in the Picard
iteration. Then during the calculation of element values at
the Gauss points (velocity, velocity gradient, stress gradients,
etc.), the boundary elastic stress at the corner node which
34
matches the Gauss point is calculated according to:
N.P. G.P.++_ y G.P.le 2 e R ITI
G.P. IG.P.
(IV.23)
Where N.P. is the node point and G.P. is the Gauss point.
This value of elastic stress is then used in the central differ-
ence calculation at the Gauss point if the element is on a
boundary. Figure 5 shows the details for the calculation at the
Gauss points for both boundary and interior elements as described
above.
To keep our thoughts clear, it is instructive to pause
and review. The creeping flow finite element equation to be
solved is,:
f f (BT D B + (mTk)Ta m B)dv NTLT edv NT tdA
v v A
The coefficients of u are linear and ae is solved by successive
substitution for each value of u. Notice two things. First,
NTL T =B T so that we could make this substitution. This study,
however, included the terms Vae in the equation and so theseTivalues were used directly with NT in calculating the integral.
Second, a nested iteration on ae is really not necessary.
Rather we could calculate a new u for each update of ae and
combine the two iterations. Figure 6 shows the two different
schemes. While not mathematically demonstrated, it was felt
that such a scheme would further degrade convergence since
35
would undergo much larger variations. This issue should be
considered in much more depth in continuing studies. This
section will be concluded with a discussion of three topics,
two very important, one included only for completeness. These
topics are: convergence of the solutions, simplication due to
ignoring the stress gradient terms of the constitutive equation,
and equations used for independently interpolating the total
deviatoric stresses in a mixed finite element method. We
will discuss these topics in order.
Engelman et. al. (32] consider the problem of convergence
of the general Navier-Stokes equation noting that Picard itera-
tion converges more slowly than Newton-Raphson, but normally
over a larger radius. They then treat the issue of accelerating
convergence by employing guasi-Newton methods emphasizing
Broyden-Fletcher-Goldfarb-Shanno updating. Such acceleration
methods would enlarge the number of elements which can be
economically treated in the solution scheme. Currently, however,
this is not the problem with viscoelastic flow. As we will dis-
cuss in Section VI, the radius of convergence is the major issue,
not the rate of convergence. Our study succeeded in obtaining
solutions for Ws<0.01 which could possibly be considered a trivial
case. However, for the general flow geometries, we treated (in
particular entry flow), the studies cited in the beginning of this
section failed to achieve solutions even at that limit. Con-
vergence therefore is the critical barrier to obtaining more
general viscoelastic solutions. We did not pursue such extensions
36
in this study, but it is worthwhile to suggest a possible
approach. Chung's [2] review of standard solution techniques
is directly to this point. The radius of convergence can be
widened by continuation methods. In particular, Chung suggests
a multiple solution technique which combines incremental load-
ing with Newton-Raphson corrections. Future effort in this
field should investigate such an approach. We employed Picard
iteration exclusively. Picard iteration should be tried as the
top level, along with continuation methods. It is noted that
both types of solution are amenable to the computer program
used in this study.
We turn now to the simplications when the stress gradient
terms are neglected. The terms themselves arise in the con-
vection terms of the constitutive equation, i.e., (u4V)a. For
creeping flow similar terms were neglected in the Navier-Stokes
eq-lation and we know that for polymer melts, this is a good
approximation. It is then obvious that we compare approximate
magnitudes of Vu and Va. For viscoelastic flows, we have already
established that a eis on the order of av and the gradients
might be expected to be of equivalent nature. Therefore, we
look at the comparison between the first derivative of u and
the second derivative. It is known that even when u is discon-
tinuous (as in the case of cross-channel flow of a screw extruder
(9], the approximation at small distances from the singularities
of Vu are quite good. This suggests that for creeping flow, a
good approximation may be achieved when (u.V)a is neglected.
37
. I .Aim
Equation IV.13 then becomes:
ae = vA(&V + _e) (IV.24)
Evaluation of Vae is eliminated and the Picard iteration be-
comes much more straightforward. An optional approach is to
solve oe explicitly as:
X_)e - XA ov (IV.2s)
or
_e. - AA)- XA a (IV.26)
Where I is the unit matrix 6ij = 'j (ij = 1,2,3)
Equation IV.Y6 allows IV.5 to be written explicitly in terms
of u and the equation is a simple non-linear equation which
can be solved with the numerical techniques discussed throughout
the report. It is noted that although the explicit form makes
the equations more straightforward, it is not expected that
the radius of convergence (which is a function of X) will be
widened much. However, at the early stages of research efforts,
particularly in applying continuation methods, this equation
seems to offer promise.
Finally, the mixed method of solution is briefly discussed
for sake of completeness. Following Kawahara's approach [28],
we set up the simultaneous equations for steady state in indicial
notation:
38
- -- - - . .. . . ...
p u ui, j + P,i - ij,j =o (IV.27a)
Iij + X(ukoij,k - Ui,kOkj uj,koik) - 1(ui, j + uj i ) = 0
(IV. 27b)
Both IV.27a and IV.27b are non-linear; we write the finite
element equations:(IV. 28a)
I(TP(V_(n U)T) TN + (T _T T B *T~vif(u)T (mr a)m ) dv U+ jf *7,dv a-- fV V
ff*T _A *T DB- 2Nd~ +{/jT~d~~XV(N a)N - N D B -N Q )dv u+ N a = 0
V
(IV.28b)
Where the asterisk indicates the trial function for the
stress interpolation.
The solution to IV.28 can be seen clearly if we form a typical
equation in matrix form:
T ATT ANTp(V.(Nu) T) TN. F1
+ (m B.) am B. N.B v l-- i - 2 (IV.29)
L--------------------------------...... I,
N. XV(N N
S*TA-N. DB I N. N. aY
1 = yy
i - -N ETNj _xy
(In equations IV.29, the integrals are implied.)
39
In IV.29, the superscript in the column vectors indicate the
node number so that this relation is repeated for each of
the nine nodes. i and j indicate the row and column in the
assembled array (for IV.29 i=j=l). The array is partitioned
accordingly so that the upper left corner is 2 x 2, upper right
corner is 2 x 3, lower left corner is 3 x 2, lower right corner
is 3 x 3. All matrices in IV.29 have been previously defined
with the exception of which is:
2(N X x + 2(1 xy ) )- 0
* A~*^ a *^ a *^ *^ aNCyy + Naxy + ay
Ty- xy x -xx ax - xy ay
These equations when fully assembled yield a set of linear
equations of order 5p, where p is the number of nodes. For a
nine node element then the order of equations is 45. The number
of variables for the whole domain then would be 45n-m with n
being the number of elements and m the number of shared nodes.
It can be seen that it does not take many elements to generate
a very large computer region to solve the equations. While the
above analysis was conducted and subroutine ELMT06 written for
the problem solution, no flow cases were run in this study.
Future work may implement subroutine ELMT06.
40
V. COMPUTER IMPLEMENTATION
In this section we will discuss the major aspects of the
finite element program, the calculational procedures, and the
input/output.
The source program was a modified version of the Finite
Element Analysis Program (FEAP) written by Prof. R. L. Taylor
at the University of California, Berkeley, and published in
Chapter 24 of [1]. The modifications have been made by
Prof. David Roylance of the Massachusetts Institute of Tech-
nology to accommodate polymer melt flow [9]. These modifica-
tions are largely: (i) addition of a power law flow rule,
(ii) addition of a temperature dependent viscosity, (iii)
alteration of matrix algebra operations, and (iv) addition
of an axisymmetric capability. The rationale for using this
model is given in [9]. The current effort included reviewing
the source program to insure correctness, and modifying it
to include a viscoelastic flow option. Currently the program
is two-dimensional (rectilinear or axisymmetric) and steady
state.
The program establishes a dynamic storage vector at the
outset which is partitioned to store all input data (node co-
ordinates, element node numbers, etc.), global data (stiffnesses,
loads, etc.) and output data (velocities). Other features are
a linear interpolation mesh generation scheme, an active
equation solver and a macro command language which controls
the solution execution. The macro commands and their meaning
41
are listed in Table 24.12 of [1].
Upon construction of the architecture of the problem,
calculations required for a specific command (such as forming
the tangent stiffness matrix) are made in a library of element
subroutines. Subroutine PFORM steps through the n elements by
forming element arrays from global data and passing the arrays
to the element routine. Subroutine ELMT05 is a general 2D
penalty method solution of the Navier-Stokes equation written
by Frecaut [16). This is the element subroutine modified for
the viscoelastic flow.
The basic source program flow chart is given in Figure 7.
To modify this program for viscoelastic flow, three basic
changes were made. First was to flag the problem as visco-
elastic and read material data. This was done in subroutine
DFMTRX. The card reading format after input macro command
MATE was changed to the following:
CARD 1 Format (15, 4X, Ii, 17A4)
CARD 2 Format (415, F10.0)
CARD 3 Format (15, 7Dlg.4)
Card one reads the material set number in columns 1-5 (in all
cases only one material set is used and therefore this is 1),
the element type in column 10 (5 for ELMT05) and the problem
description in the remaining columns. Card two reads the flow
type in columns 1-5 (1 = plane flow, 3 = axiqymmetric flow),
a flag (Ni) for thermal coupling in columns 6-10 (0 = isothermal,
1 = thermally coupled), a flag (K2) for viscoelasticity in
42
columns 11-15 (1 = simple viscous, 2 = power law viscous,
3 = White-Metzner Viscoelastic, 4 = DeWitt Viscoelastic,
5 = Rivlin Ericksen Viscoelastic), a flag (N3) for the time
domain in columns 16-20 (1 = steady state, 2 = unsteady),
and the power law coefficient (P4) in columns 21-30. P4
must be included and for simple viscous material P4 = 1.0
(which was the case treated exclusively in this study).
Card three reads the Gauss integration order (L) in
columns 1-5 (2 = 2x2), the penalty coefficient (XLAM) in
columns 6-15, the viscosity coefficient (XMU) in columns
16-25, the density (RHO) in columns 26-35, the viscoelastic
shear modulus (G) in columns 36-45, the thermal conductivity
(XK) in columns 46-55, the specific heat (C) in columns
56-65, and the work-to-heat conversion factor (HEAT) in
columns 66-75. The program is written so that when data is
not required for the specific problem (e.g. linear, steady,
isothermal, inelastic flow) those columns may be left blank.
In card three then only columns 1-25 need be included.
The second change was to add algorithms in ELMT05 for
the calculation of the elastic stresses according to equations
IV.13. The last change presented the major difficulty: the
calculation of the elastic stress gradients according to
equations IV.19. As noted in the previous section, no scheme
existed for making calculations with variables from different
elements. In order to solve IV.19, however, this was necessary.
The approach taken was to define common arrays YY(I,J,N), ESIGl(I,J,N)
43
ESIG2(I,J,N),ESIG3(I,J,N),ELASl(I,J,N),ELAS2(I,J,N),ELAS3(I,J,N),
and BOSIG(I,J,N). YY is the global coordinate (J=l,2) of the
Gauss points (I=1,4). ESIGI, ESIG2, and ESIG3 are xx' yy
and axy respectively at the Gauss points (I=1,4) at iteration
J=K, K+l. ELASI, ELAS2, and ELAS3 are the gradients (J=l,2)
of ae at the Gauss points (I=1,4). BOSIG is the elastic stress
(J=1,3) at the boundary, at the corner node (1=1,4). In all
the arrays N is the element number. In PFORM, N is passed as
common through ELMLIB and ELMT05 and it is therefore possible
to conduct the calculations between the two subroutines PFORM
and ELMT05. The gradients of the three stress components at
the Gauss points are first solved for all the elements assuming
they are a boundary element on all sides. A searching scheme
is then affected which compares the nodes of all the other
elements. When two elements are found in the correct location,
the elastic stress gradients are replaced at that Gauss point.
If adjacent elements are not found, the element is on a boun-
dary and that Gauss point is left unchanged. During the
Richardson/Jacobi iteration, the elastic stress gradients then
are calculated in PFORM and these values used in ELMT05 to
calculate the updated values of the elastic stress at the
K+l iteration. This iteration is conducted 20 times unless
convergence is achieved beforehand. The program then continues
in a normal manner.
The listings of the major subroutines written to accomplish
the modifications are included in Appendix 3. The subroutines
are in order listed ELMT05, ELMT06, ESHAP, PFORM, CMATRX, and
44
FPSIG. ELMT06 is the subroutine written for interpolate total
deviatoric stresses in a mixed method. ESHAP is the calcula-
tion of the second derivatives and FPSIG is a new routine
written to print viscous and elastic stresses at the Gauss
points. CMATRX is the subroutine which forms the Q matrix
in ELMT06.
45
-. -.--- ' ~I
VI. CALCULATION RESULTS
Four flow geometries were treated as shown in Figure 8
(along with the boundary conditions): Cross Channel Flow,
Plane Couette Flow, Entry Flow, and Step Flow. Table 1 shows
the computer run matrix. The input data sets for runs 1, 3, 4,
6, 13, and 20 are included as Appendix 4. Results are dis-
cussed below for each of the four problems treated. For all
cases, the viscosity coefficient was taken to be 790 poise.
This was the value selected by Roylance [9] in previous studies.
His reasons were unrelated to the work in this study, but we
chose to use the same value for comparison purposes. With
more reasonable values (10 4), we would only expect to see
higher stresses, but no change in the velocity fields.
CROSS CHANNEL FLOW
The solution of creeping flow, circulating in the trans-
verse plane of channel, for a viscous fluid is well known (e.g.
[9]). At steady state, the circulation is uniform with a vortex
center at mid-height, towards the vertical boundary on the right
in Figure 8a. This study looked at the consistency of repro-
ducing this flow with 9 node and 8 node elements and the effects
of a finite fluid elasticity. Secondary eddies and screw power
requirement changes were considered to be demonstrable effects
of elasticity.
46
.. . .. . . . . . . . ." - - !--
Figure 9 shows the velocity vector flow field for run 1
(linear case). Results are identical to [7], different from [9).
This is due exclusively to the specified boundary condition at
the upper corners of the channel. For our boundary .onditions,
the vortex center is at the mid-width of the channel near the
2/3 height section.
The velocities calculated for the nodes of elements 7, 9,
and 15 by the 18 element 9 node and 18 element 8 node case
are compared in Figure 10. Note that a significant difference
occurs in the direction of the resultant velocities in element 7
and the magnitude in element 15 (a 20% lower horizontal velocity
is predicted in the middle nodes of element 15 by the 8 node
model). When the results of the 72 element, 8 node case are
examined (run 3) the 9 node model is found to be uniformly
closer. The velocity field is, therefore, predicted much better
by the 9 node elements for the same number of elements.
Let us now make a practical application. The power per
unit area required of a single flight screw extruder to create
this circulation is the shear stress in the fluid times UB
(the relative barral velocity). If we approximate this as the
average element shear stress Uxy times the average velocity
in the element, we have the following for element 15:
47
9 NODE 18 ELEM 8 NODE 18 ELEM
266Sxy(dynes/cm2) 0.22 x 106 0.2 x 106
(cm/sec) -50 -52.5
(dyne-cm/cm2-sec) 1.08 x 107 1.05 x 107
We can conclude that the 9 node elements yield more accurate
node velocities, but when average properties are sought, such
as the power or torque required for the screw design, both
models give approximately the same results for equivalent
meshes. This, of course, is expected since the finite element
equations satisfy equilibrium over the entire region. However,
on a local scale (which we are also interested in) the above
justifies our earlier preference for the 9 node elements.
From Hughes data [7], the effects of increasing the
Reynold's number (Re) is to shift the vortex center toward the
right-hand boundary. This was investigated for one case by3
choosing the density of polyphenelenesulfide (1.6 gm/cm3).
Combining this with the other characteristic numbers of the
cross-channel flow problem, we obtain Re = UL = 0.41.
Including the convection non-linearity for this Re we found no
discernible perturbation to the velocities or stresses, thus
confirming the validity of the "creeping flow" analysis.
For the single viscoelastic case for which the solution
converged (Ws = 0.02) the velocity field again did not vary
appreciably. Figure 9 can, therefore, be considered correct
for this level of elasticity. To look at the stress effects,
48
we make the same calculation for the specific power as above
yielding:NEWTONIAN VISCOELASTIC (Ws=0.02)
a 3y (dynes/cm2 0.22 x 106 0.22 x 106
u (cm/sec) -50 -50
27w (dyne-cm/cm -sec) 1.08 x 10 1.08 x
Within roundoff error, the two flows are identical (maximum
a deviation was 1%). A second comparison is available inxy
Figure 11 where the pressure is plotted at the mid-height as
a function of the cross-channel (transverse) station. Again
the viscoelastic flow is coincident with the Newtonian case.
Within the range of calculations achieved in this study
therefore (Ws<0.02), there are no effects of viscoelasticity
manifested. We do observe, however, that the stresses calcu-
lated (-l% variation) are consistent with the Ws suggesting
accuracy of the computer model when convergence is achieved.
PLANE COUETTE FLOW
Plane Couette flow was selected for the fundamental
evaluation of the computer model. This is through the relation
presented by Middleman [26):
SR = (VI.l)
where S R is the recoverable (elastic) shear stress:
SRa xx 0YR 2axy , (VI.2)
49
X is the relaxation modulus and y is the steady, simple
shear flow strain rate. The flow is enforced by specifying a
linear variation of the horizontal velocity between two plates,
one stationary, the other moving at a constant velocity as shown
in Figure 7b.
Run 7 was the Newtonian case to validate the problem. In
this case, axx and ayy should be identically zero and xy
constant throughout the field domain. This was the result of
the calculation.
For the viscoelastic case (Run 10), all the normal stresses
are elastic while from equations IV.13, with v = - = o only
a e is finite. Therefore, we should observe the following:xx
e
SR Xx = Xy- constant (VI.3)xy
For a unit height between sliding plates we have y = UB SO
that:
a xxe = 2UB a xv (VI.4)
The computer results are for X 0.0002, UB = 100 cm/sec
(Ws = 0.02):
e = 3.16 KPa, 2XU Bayv = 3.16 KPa.OxxBy
The equation is identically satisfied. This, of course, is
encouraging for future work to increase the radius of convergence
for higher Ws numbers.
50
ENTRY FLOW
The entry flow problem for viscoelastic fluids has not
been successfully calculated by finite element methods in
the past, due to severe numerical convergence problems. As a
first step, Run 11 was accomplished for linear flow according
to the boundary conditions specified in Figure 7c. A discus-
sion of these boundary conditions is in order.
Rather than a constant horizontal velocity at the inlet
to the reservoir (upstream channel), a more accurate analysis
would specify fully developed flow. Middleman [26] presents
this for flow between parallel plates (for a Newtonian fluid
as):
B2 = )-2] (VI.5)
where B is the channel height
and L is the channel length
(all other variables retain their earlier definition).
For a White-Metzner fluid, the plane-Poiseville flow
would be solved by adding the elastic stresses to the momentum
equations. Perera [27) did this for a 4 constant Oldroyd fluid
and solved the resulting second-order differential equation for
u(y) by Newton-Cotes integration. With equations of the typeaP
specified in VI.5, we can solve the pressure loss 3- due to
inlet and outlet. In addition White [33] cites the additional
pressure losses due to entrance and exit of the dies. It is
these boundary conditions that would be more realistic in
treating the entry flow problem (velocity according to VI.5
51
at one end, AP at the other). With the formulation specified
in this work, it was expected that the flow field would behave
quite differently from the classical converging type. Since
we did not have data on die pressure losses, however, the
initial calculations were made on the basis of the boundary
conditions given.
When fully developed conditions are specified, both upstream
and downstream of the entrance region the flow is known to be
stable up to relatively high Ws numbers. At Ws around one
secondary vortex patterns arise which are generally ascribed
to increasing elastic stresses generated in the shearing/elonga-
tional flow (White [33] implies that elongational flow is im-
portant and we, therefore, conclude that the Rivlin-Ericksen
fluid simplified for viscometric flow is a questionable model).
This flow behavior is documented in Figure 12 which shows
experimental behavior noted by White [33) as a function of Ws
and calculations of Perera (27) for Ws = 0.6.
The calculated velocity field for the boundary condition
specified in Figure 8c is shown in Figure 13. Although the
mesh is very coarse, it appears that the flow is unstable for
these conditions. The viscoelastic calculation (Run 12, Ws =
0.01) exhibited identical behavior. Because of this poorly
behaved flow field, the calculation was repeated using the
fully developed flow boundary conditions. The results are
shown in Figure 14. The specific boundary conditions were
established in the following manner. The excess pressure
losses described by White [33) were ignored (this will affect
52
I-A
the calculation however). At y - 0 (y measured from the
mid-height of the channel) equation IV.5 is
B2p8--- (IV.6)
For the two channels, there would be a total pressure loss of
APT = API + AP if the flow was fully developed. Therefore
APT =8U --- + ----- (IV.7)B1 Bo
For our geometry LI = L° = BI = 1, Bo= , and V = 790 poise.
There are three unknowns in equation IV.7. However, rather
than specifying two of the three, we merely let API = AP0
(which has the same effect) and specified uo . This permitsthe calculation of uI and thereby calculation of AP This
APT was established as the inlet traction PI and the outlet
was atmospheric Po = 0. This yields the value of PI = 6.4j.
The pressure is converted to the virtual "work" equivalent
node forces by the relationship
Fc = H PIx 3
- _ H P
x 3 I (IV.8)
where the superscript denotes the element node (c - corner
node, M = mid-side node) and H is half the element height at
the nodes. (See Precault [16] for the details of virtual "work"
equivalence calculat*ons; equation IV.8 are valid for 8 node
and 9 node plane quadrilateral elements). (In the actual
boundary node forces, the corner nodes are loaded with
53
....... ......
c 2FX - H P, for uniform meshes since the node is shared by
adjacent elements. Only the vertical velocities at the
boundaries (v=o) now must be specified. The mid-height hori-
zontal velocity will not be the value used in the calculation,
but the flow will be fully developed.
Comparing Figures 13 and 14, we see that although the
behavior is somewhat improved by the fully developed flow case,
there is still major error in the flow field and even flow
reversal. This is felt to be attributable entirely to the
coarseness of the mesh, particularly near the entry corner.
A finer mesh case was not constructed to test this hypothesis.
It is recommended that future work include this refinement.
Notice that symmetry was not employed to reduce the
number of elements. This was due to the difficulty of speci-
fying boundary conditions on the plane of symmetry. The first
conditon is v=o, but the other boundary condition is not so
straightforward. We know that D and a- are zero at the mid-av
height, but in general w is not zero within the reduction
region. This, of course, is a statement that the one dimen-
sional lubrication theory is not valid. Since the pressure
now changes across the channel height the pressure in the x
direction can no longer be specified as a linear function of x.
Therefore, the nodal loading in the x direction is unprescribed
as well as the velocity u. Of course, this could be resolved
by adding the condition of no mass flow across the plane of
symmetry. We chose not to accomplish this at the penalty of
doubling the number of elements.
54
If the flow were one dimensional, the pressure would vary
linearly with the length and the velocity would be constant
in each of the two sections (plane Poiseulle flow). Figure 15
shows the deviation from this case.
It was noted in examining the stresses in Runs 11 and 12
that the difference was much larger than expected for the low
Ws. However, a thorough evaluation was not conducted because
it was felt that the differences were an artifact of the
calculations due to the following: (i) the velocity fields
were erratic as previously mentioned due to the coarse grid,
(ii) the boundary conditions of constant inlet velocity gave
rise to poorly behaved pressure variations even for the
Newtonian case, and (iii) the solution convergetce for the
non-linear problem was still poor at 30 iterations. It is
noted in passing, however, that as the solution procedure is
improved, it is exactly these types of variations which are
being sought.
STEP FLOW
This geometry was selected as the beginning step toward
an analysis of flow past an obstruction such as would be the
case if pins were added to the cavity to form holes in the
molded part. With the boundary conditions specified in
Figure 8d, the results were very similar to those discussed
for entry flow. A discussion of the computer calculations
will therefore not be included in this report. It is noted,
however, that there is still negligible differences between
Runs 15 (linear Newtonian) and 16 (convection Newtonian).
55
This begins to address the issue of "Stokes paradox" and the
necessity of including convection, even for low Re, for
obstructed flow. The paradox is that in two-dimensional
flow no analytic solution exists for the linear equation
which matches the boundary conditions at the surface of the
obstruction and at large distances away from it [34].
Batchelor [35] shows that when the distance from the obstruc-
tion (or a boundary) is on the order of I/Re (where I is a
characteristic dimension of the obstruction) the convection
stresses (inertia) may become of equal importance to the
viscous stresses. Analytically this correction is known as
o'seen's improvement. Again as the model described in this
report is refined, the adequacy of the "creeping" flow
analysis must be examined in light of this issue.
56
I I .
VII. MODEL EVALUATION
It is worthwhile to complete a qualitative evaluation
of the computer model before this report is concluded.
Figure 16 presents a diagram of a complete model for a real
injection molding process. The Figure emphasizes those
elements included in this study. Since we achieved numerical
convergence for Ws < 0.01 it must be concluded that a non-
Newtonian power law fluid analysis would be as good an
approximation as the viscoelastic model used herein. If
future work does not improve this convergence region (at
least to Ws > 0.5) the numerical analysis would seem to be
as good without including elastic effects. Also finite
difference methods have succeeded in obtaining solutions up
to Ws = 0.6 [27] and it may therefore be advisable to develop
these techniques for application to the gyroscope manufacturing.
The model is steady state and includes no free surfaces
such as would occur during the mold filling period. Therefore,
it can only be used in regions such as the extruder, nozzle,
sprue, runner and gate. Unless unsteady, free surface terms
are included, this model is not applicable to the mold filling
itself. But the power required to supply a nozzle with a given
rate of flow is certainly within the capability of the model.
Also the state of the bulk material as it passes through the
gate can be determined by use of this model. Any damage due
to high stresses or thermal degradation in these regions can
be analyzed with the model. It is noted that although there
57
.------.---.-.------
will be a finite elastic stress which the polymer can sustain
before flowing completely plastic (viscous plus the elastic
limit stress), there is no yield stress built into this model.
Therefore, while the model will predict continually increasing
stresses, judgement must be exercised as to the real elastic
capacity of the fluid.
The current status of the coupled heat transfer capability
of the model is the adiabatic model developed by Roylance [9].
Extension to a complete non-isothermal boundary analysis can
be implemented without too much difficulty.
We have noted that major modifications are necessary to
evaluate the mold filling itself (only pressure and filling
rate can currently be analyzed). Also within the mold, the
cooling stage of the molding process can not be analyzed
because of the absence of a solid thermomechanical viscoelastic
model.
However, if an initial state can be established for the
cooling process such a model could be developed.
The mold filling process itself can take the approach of
a constant flow rate at the gate once free surface effects are
added to the model. This is the approach used in [11]. The
free surface analysis is most clearly discussed in [4] where
the front displacement is calculated over some interval of
time assuming a constant velocity of the boundary elements
node points. The surface traction on the flow front is zero
normal to the surface and the material surface tension tangen-
tial to the surface.
58
From Section V, we can discuss the approach to improv-
ing the viscoelastic case. To assess the maximum radius of
convergence of the momentum equation, it is adequate to
neglect u.Va and use equation IV.26. Since Newton-Raphson
iteration generally converges for the Navier-Stokes equations
well above Re = 25, it should be verified that convergence
is achieved with the current numerical approach for Ws = 25.
With this step accomplished, u-Va can be added and the con-
tinuation method used. The effective technique should employ
incremental loading with Newton-Raphson corrections. Let
us discuss this a little further. Since we are using direct
(Picard) iteration on the elastic stress terms, let us rewrite
equation IV.ll as:
Ku = f - Ke (VII.l)
Since Picard iteration is a single point scheme, (i.e.,
the initial value of K u + Ke- f is always used rather than
updating in the Newton-Raphson scheme, see Figure 17) we can
attempt to increment this point. Therefore, instead of solving
VII.l directly, we solve:
Ku = 9(f - Ke ) (VII.2)
where O<e<l. With the solution to VII.2 converging for
sufficiently small numbers of e we can update the initial se-A
lection of u by incrementing e. For example, let e = 0.1 then
in the first inurement the first value of u is
0o = K-15 K ef
* 59
. .*
We then iterate with K u = el - Ke(us)].
When convergence is achieved then we increment 8 to 2e - 0.2.
Then 0 1 1K -[ 2eF - ho (u s + ' ) ]
Therefore, the initial guess is improved by the correction
Ke (u +). It is noted that this technique is different from
the normal continuation methods where the non-linear equation
is always of the form: K(u)u=f. While no mathematical analy-
sis has been conducted on this proposed technique, it appears
to offer promise.
This deviation in the classical incremental load method
is only necessary when the stress gradient terms are included
in the viscoelastic constitutive model. Therefore, when the
model undergoes its first revision with u.Va neglected we
write 0e explicitly and if convergence fails the classical
incremental load methods described in [1] and (2] should be
employed.
60
VIII. CONCLUSIONS
This report has dealt primarily with the additional
mathematics required to incorporate elasticity in the
deviatoric stresses developed in flowing polymer melts.
Implementation of the equations within an existing Finite
Element Computer routine was then shown. From these analy-
ses we can make the following conclusions:
* The direct Picard Iteration Converges within a
radius of Ws<0.01.
e For cross channel flow and entry flow "creeping"
solutions are very accurate for typical polymer extrusion
Reynold's numbers (Re<0.4).
* For the Weissenberg numbers which yielded conver-
gence, no appreciable effects on the flow were noted.
e The programming technique of passing data between
elements by common memory appeared to be effective.
* When convergence was achieved, the calculated values
of elastic stresses were consistent and reasonable.
* The penalty method of incompressible flow appears
to yield good results for viscoelastic fluids.
e The radius of convergence was consistent with previous
finite element calculations.
* The radius of convergence can certainly be improved
by finite difference calculations as evidenced by Perera [27).
* Without improvement, the only computer options which
should be used in evaluating polymer fluids are Newtonian and
power law viscous (isothermal and adiabatic).
61
e Techniques of improving the viscoelastic model have
been proposed which offer great potential.
* For 24 element problems, the computer cost for runs
requiring 30 iterations was $100.00.
62
IX. RECOMMENDATIONS
It is felt that the work performed in this study offers
potential for useful follow-on effort. In particular, there
are three areas of development. First, the analysis of the
complete flow problem is vital. While the gyroscope fabrica-
tion is new, the need for numerical evaluation in the molding
process is not. The work at Cornell [11] demonstrates this
fact. In that effort, the various regions of flow are being
tied together. A similar approach is required for the finite
element modeling. A model which connects the flow within
and out of the extruder, through the various conduits, and
into the mold cavity is an important development which should
be pursued.
Direct extensions of the work addressed in this study
are also important. The approach should be: (i) ignore
stress gradients in the constitutive equation and conduct
direct calculations, (ii) add stress gradients along with
continuation solution methods of non-linear equations. Even
if future work with constitutive models which include stress
gradients are unsuccessful, it is felt that the equation
with some elastic stresses will be a big improvement over
Newtonian or power law fluids.
63
Finally as efforts one and two above progress, there is
a need to conduct rheology experiments which will determine
properties of the fiber-filled polymers being used in the
gyroscope fabrication. These data are required to correlate
with the velocities and stresses predicted by finite element
equations.
The three categories are listed below:
e Model complete flow history from extruder to
mold cavity.
e Refine viscoelastic model.
e Conduct rheology experiments of appropriate polymeric
materials.
64
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U
___ 00.
z .r50
ILl2z
LU rC - .4J
w0
zr-4
LU L
>. IC02 t 4
D 2r
UjL
66
Figure 3 Fluid Maxwell Element
67
COL.i ICOL. i COL. i+ I COL. i+ 2
ROW ( + 1)
ROW I i -1 j) 0 (0.1)0 +
ROW i- 1 *(i, j -1)
ROW i-2
x
Figure 4 Non-Uniform Molecule Mesh for Solving StressGradients
68
BOUNDARY
O NODES0 GAUSS POINTS
EQUATION IV-18 USED AT G"JEQUATION IV-23 USED TO CALCULATE aN.P A-e ATEQUATION IV-19 MODIFIED AT ('jlAS FOLLOWS:
Xijl= X(®, ij+i = Y(Z® 0 i,j+i = 0@
(Note: Superscripts are indexed at each Gauss point so that x(Dis x j+ referred to Gauss point 1
whereas itis x1 ,j+1 referred to Gauss point 4)
Figure 5 Calculation of Elastic Stresses at Gauss Points
69
SI ASOLVE u FOR SOLVE . FOR
CREEPING INELASTIC CREEPING INELASTICFLOW FLOW
k+1 SOLVE k+1
SLEoSOLVE ^so
U 4
TO
SOLUTN
•NO ACOMPLETE
SOLOLUTION
. COMPLETE
(a) (b)
Figure 6 Iteration Schemes for the Solution of Creeping,Viscoelastic Finite Element Equations: (a) NestedIteration (b) Combined Iteration
70
____________- -*-.-.*--- v--
MAIN i• ESTABLISH BLOCK
FEAP COMMON FOR DYNAMICSTORAGE VECTOR
STOP 0 CONTROL PROGRAM
PCONTR EXECUTION
PMESH READ INPUT DATAAND GENERATE MESH
*ESTABLISH PROFILPROFIL OF EQUATIONS FOR
SOLUTION
END * STEP THROUGHPMACR MACRO COMMANDS
.* SOLVE UNSYMMETRICEQUATIONS
* FORM ELEMENT ARRAYSPFORM FROM GLOBAL DATA
PRTDIS *PRINT V LOCITIES
ELMLIB 0 CALL DESIRED ELMT
o FORM TANGENT STIFFNESSE LMT05 MATRIX/FORM OUT-OF-
BALANCE VECTOR/CALCULATE AND PRINTSTRESSES
Figure 7 FEAP Flowchart
71
__ ..- . . -
y 1; u = -100cm/sv =0
(CORNER B.C.)
x=2;u v = 0
x = 0; U=V 0(a)
y = 2; u --00cm/ v = 0
x =0; U= -100y, 2=- O10v
(b)
y=1; U =v =0.= V< 0 2 5
v1 > y > 0.75 uv 0
x = 2;x =0; u = 100cm/s, v =0 FULLY DEVELOPED FLOW
v = 0, P+Oxx =0
y = 0.25,0.75 U 0
•/~~~ ~ I; = O- ~ v 1 0(c)
II
x- x 2;
x =0; u = 100cm/s, v = 0 FULLY DEVELOPED FLOWv = 0, P+uxx = 0
LOWER BOUNDARY; u = v = 0
(d)
Figure 8 Flow Geometries and Boundary Conditions:(a) Cross Channel Flow (b) Plane Couette Flow(c) Entry Flow (d) Step Flow
72
. .
U
0 Z)-
HI .-
U0
H 0*-4 -0
0 1 J-W
:3
IN tv
73
,Oi 5 0 c m
58=10 cm/s
J I v , c~I II I
(a Elmn 15 (b(amet9( )Elmn7
I II II I
I 50 cm/s
I II I
I I
I I.I I
1, I 50 cm/s
I
(aI lmet1 (b Elmn c)mn7
(Dashed Arrows are 8 Node Elements)
74
0r
ci-4 ...4
. 0
m090 (0 0 o
-P 04 c
u 0
.LI 0
m 10
o Uo
-n 0 II
0)oz W*~*I >
dGi4Jd
75~
(a)
80
70-6050
S 40S30S2010-0-
1 i I A I a &oli1.0 10
ws
(b)
SM(c)
Figure 12 Fully Developed Flow Behavior of ViscoelasticFluid Entering and Leaving a Contracting Channel:
(a) Vortex angle 6 (after White [331) (b) B vs Ws(after White [33]) (c) Finite Difference Calculation
for Ws=0.6 (after Perera [27])
76
- ~- -.--- i-
0..
4
41 t
04-I-*d4 0> 4 -
(V -U 4 0 I
0 to 0U 4.0 - 4
0 14 4
77
00
*d 0
41.
04J*4 0 W
* 0~ 0
r4 0
C IV)
0~ 0
r- 4J
0~ U
0
A78
(d)) d
xx
0 4 )
4 .
$4 -4'C 0
I44 M2
x -4
'C 00
0 $4>
0 4-4
w >
>2
0 (
>0>lox II0U T
xx
0
'4C
0 0 0 0D e m
01L X s/UJo) nl
79
~ 0 -0
20 z 0 0
LL CLU.
wi z 3 <- XC z -j
00
0
a4 4
0 4
-1 r
9 1 4-) >4L z U, 4)
0 1
z8
-p-Rpm
Ku Ku
uO u1 u2 U 3 U0 U1 U2 U3
(a) (b)
Figure 17 Solutions to Nonlinear Equations (a) Picard
Iteration (b) Newton-Raphson Iteration
81
0
JI Iuj 0 X w
LUU
0 (J 0
u.jLUU
zz
CD CC0 0 0u
'UU0. U0
824
- . S S .4
CONVERGENCERUN NO. GEOMETRY TYPE (YES OR NO) COST ($)
1 CROSS CHANNEL LINEAR EM2.0018-9 NODE ELEM.
2 CROSS CHANNEL LINEAR 2.0018-8 NODE ELEM.
3 CROSS CHANNEL LINEAR 3.0072-8 NODE ELEM.
4 CROSS CHANNEL CONVECTION (Re 0.4) YES 3.5018-9 NODE ELEM.
5 CROSS CHANNEL VISCOELASTIC (WS = 0.1) NO 10.0018-9 NODE ELEM.
6 CROSS CHANNEL VISCOELASTIC (WS = 0.02) YES 21.3518-9 NODE ELEM.
7 CROSS CHANNEL VISCOELASTIC (WS = 0.06) NO 32.9618-9 NODE ELEM.
8 PLANE COUETTE LINEAR 2.0018-9 NODE ELEM.
9 PLANE COUETTE VISCOELASTIC (WS = 0.06) NO 12.4718-9 NODE ELEM.
10 PLANE COUETTE VISCOELASTIC (WS = 0.02) YES 23.4118-9 NODE ELEM.
11 ENTRY LINEAR 2.0024-9 NODE ELEM.
12 ENTRY VISCOELASTIC (WS = 0.01) TENDING AT 87.7724-9 NODE ELEM. 30 ITERATIONS
13 ENTRY VISCOELASTIC (WS = 0.001) YES 77.0024-9 NODE ELEM.
14 ENTRY VISCOELASTIC (WS = 0.03) NO 37.3924-9 NODE ELEM.
15 STEP LINEAR 2.5030-9 NODE ELEM.
16 STEP CONVECTION (Re - 0.4) YES 21.4630-9 NODE ELEM.
17 STEP VISCOELASTIC (WS - 0.01) YES 115.0030-9 NODE ELEM.
18 STEP VISCOELASTIC (WS = 0.001) YES 79.1230-9 NODE ELEM.
19 STEP VISCOELASTIC (WS - 0.03) NO 50.7130-9 NODE ELEM.
Table Computer Run Matrix
82a
...... -----
APPENDIX 1
Derivation of Elastic Stress Gradient Expressions
From figure 3 we can write the Taylor series approximations
for Va as:
Forward Difference: ail~ a WiX + Yl Jjxf +
2
1 jijX 2 + 1 32a~i jAy2 +
0i,j+l = i~j + Da~Ax
1 92a iA*2+ 1 32a *2'T -a .Ax + 75. h +
Backward Difference: a il,) U 3*1I -a 1i Axb - Dal i,jAyb +
~~~~2 _-IAx . -- i,jAy2 +
0i,j-1 = i,j - ac - aIi,j A*
82a~ijA *2 1 *2*2 1 l~A*
ax2 ay~% i2 jAb
where:
Ax f = xil~ - x 1 ' Ayf = y1+l y1
Ax* = x ~j+l - x , Ay * = y3. ~+1 - y i,j
f Yf
Ax~ = x1 ~ 1 i-l 1 ,yb=YI -i-lui
83
-. - -- - .........
and j-i ; y1i,j " j-iAxb x x Ay -y ,
Subtracting the first and second equations of the
backward differences from the respective forward differences:
a i+l,j - i-l,j = oi i (Axf + Axb) + ( +2-0i +!i,j(AYf + Ayb ) +
1 a"a (Ax2 - ,xb,) + 1 "._i_ . (Ay2 -X2 f~ b ay 2 f~
Ay2) + ... 0(A 3 )
0i,j+l i,j- i ;(A- = li,j(Axe + Ax ) + li,j(Ay f + +
1(A -2~~ * *2 1 920A ,j f Axb ) + - - i,j(AYf
*2- Ayb ) + ... O(A3 )
Assuming that all differences of the intervals squared
are infinitesimal (zero for the uniform mesh case) and
solving for the gradients we have in matrix form:
r + Axb) (Ayf + ayb] 1 i+lj i-lj1
[(Ax; + Ax, (Ay; + Ayb [a a~'il
We can use Cramer's rule for the solution since the
determinant of the coefficients of the gradients can never
vanish. Therefore:
84
- ~ (Ayf + Ayb) - 2~+ " )(Ayf + &yb)
- (Af + &xb) CAyf + Ayb) -(&Xf Xb) (Ayf + Ayb)
=(GiI)+1 - a i'j') (AXf + Axb (a-~~ - ai- j(Ax Ax)
fAx + b) (Ayf + Ayb) - (Axf + Axb) (AYf + Ayb)
When substitutions are made for the A terms we obtain
equations IV.19.
85
APPENDIX 2
Calculation of the Global Second Derivatives
A subroutine ESHAP wab written to calculate the global
second derivatives of the velocity vector. For a nine-node
Lagragian isoparametric element the trial functions are:
N, = !(r2 - r)(s 2 - s)
N2 -- 1(r2 + r) (s 2 - s41 2 S
N 3 = I(r2 + r) (2 + s)41 22
Na = 2 (r r)(s2 + s)
Ns =-.(r 2 1)(S 2 - S)
N6 =-.(r' + r) (S2 - 1)12
N7 =-T(r - 1)(s 2 + S)
N, =-(r8 - r)(s 2 - 1)
Nq = (r 2 1) (S2 1)
We can form the following table:
86
AD-AI06 740 AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB ON F/S 20/11A FINITE ELEMENT MODEL OF A WHITE-METZNER VISCOtLASTIC POLYMER -ETC(U)
FEB 81 B R COLLINSUNCLASSIFIED AFIT-CI-I-31T L22 fflfflfllfllflfflfIIIIIIIIIIIIIIIIIIIIIIIIIIII*III ,
a2 a2 a2
ar 7S2 -rTs
NI (S2 - S) h(r 2 -r) (2r - )(2s - 1)
N2 h(s2 - s) h(r 2 + r) (2r +1) (2s - 1)
N3 '(s 2 + s) W(r2 + r) (2r + 1) (2s + 1)
N* (s 2 + s) h(r 2 - r) (2r - 1)(2s + 1)
Ns s - s2 l-r 2 r(1 -2s)
N - S2 -(r 2 + r) -s(2r + 1)
N7 -(S 2 + s) 1 -r -r(2s + 1)
No 1-s2 r -r 2 s(l- 2r)
N 9 2(S2 - 1) 2(r 2 - 1) 4rs
Writing the expressions for the second derivatives we have:
32 a a4 + 2Y a2 a ax a +~
a-as a x [Xrs y
where r,s are local coordinates and x,y are global coordinates
and the terms in brackets are merely the chain rules for form-
ing the coordinate transformations (e.g., a- = -5 - + Fa)
87
____
a~xRecognizing that terms such as and 2j are zero, we
can write the transformations in matrix form as:
a2 (ax 2 ay 2aa 2
a 2 ax x(x Bx __
a2 t~a 2 ( x
a2X2
a
* I as Bras
88
Where the Jacobianr
ax ax
has been used. All the terms in this equation are available
at the Gauss points e.g.
a2x 9 Ni
G.P. i=l G.P.
where Xi are the x coordinates of node i.
We can then solve for the global second derivatives according
to:
89
B2 ax\ 2 (a -\2 -1x\x ar) 38r/ I r 1 r
a 2 IX\ 2 2_x_-2 h ( --
2 ax a (xaT- wr. .sT3
a 2 a2r 42a - 2
- r -l
TS2 -2 as
a2 'xaE aras frs
A value is therefore returned for each of the nine trial
functions for the three global second derivatives.
90
APPENDIX 3
Listing of New Subroutines
1. EL214TO5
2. ELMT06
3. ESHAP
4. PFOIRN
*5. CMATRX
6. FPSIG
91
BRC4066 (FOREGROUND): OUTPUT FRO"I TSO XPRINT
AT 18*02:43 ON 1,4/07/80 - 5fC4066.ELlTOS.FORT
SUBROUTINE EUITOIC 0 ,UL # XL . XX . TL , S P NDF.HO"NSToIS)08000010
C 99900928
~~ ELIITOS
C v.*..**0I~*4..0000000
C A gENERAL PENALTY ELEMENT FOR ICMP-ESSIBLE FLUID PLOW 000W0070
C 490066
IPLICIT REALWO( A-N*-l 000000"
COMM 10/fLDATA/DH,A9MOTvIELoN1L 0004
COhI.0N /FVISC/IKZ 00000150
I MMC 00000165
01h!NSION 0(3O3,UL(NDF.1),XL(NWI.1),1X(13,TLI1), 00000170
1 S(tST91)vP(13,5NP(3,93,SG(9 .TG(93,RG(9 )0 0008
2 3IG(,),EPS(6),BSIG(3)PXX(3).(1)0(63)8Th813p33' co0C019c
3 5U(6),XflT5(33,XfTST(3lpPENt3 .3J,0U(3iPDLTEE(3)9 00000200
4 V(Z) ,DV(2.21 OXN(212) ,AOVEC(292) ,CADVEC(Zv2) 00000210
5 ,AlTER(3v3),ESHP(399),OOV(3,ZJ 00000215
DATA P1/3.141592653600/ 00000220
C 000002i0
if (ISW.EQ.1) So To 1 00000240
ITYPE z 0(30) 00000250
L a 0(281 00000260
RHO0 2 0(27) 00000270
XLAII 0(26) 00000,80
XIU *0(25) 00000igo
XK a 0(24) 00000300
C a 0(23) 00000310
Hl a 0(20) 00000320
HEAT_ 019) 00000330
LLB5 LS(ITYPE) 00000340
K2 0(18) 00000350
H3 x D(17) 00000360
W 016) 00000370
P2 0(15) 00000380
C 00000390
C BRANCH TO CORRECT ARRAY PROCESSOR 00000400
60 TO (29Z,3,3*5v3,3)vISN 00000420
-C 00000430
C 00000440
C 00000450
C ISM a I' READ MATERIAL PROPERTIES, DEVELOP 00000460
C OIAGOUAL-STORAGE 0 MATRIX 00000470
C 00000480
C 00000490
- 1 CALL DPHTRX(0) 00000500
- LINT 20 00000510
c 00000520
RETURN 00000530
C 00000540
92
2 RETURN 0000SSOC 00000560C cc00C570C 00000500C ISW a 3: FORM ELEMENT STIFFNESS MATRIX 00000590C 00000600C 00000610C 00000620
3 CONTINUE 00000630C 00000640C LOOP OVER GAUSS INTEGRATION POINTS 00000660C COMPUTE UNSYMI1ETRIC STIFFNESS MATRIX 00000670C 00000680
IF (L**NDM .NE. LINT) CALL PGAUSS (LgLIWSGTGiNG) 00000681DO 33 LLm1,LINT 00000690
C 00000700CALL SHAPE (SG(LL),TG(LL).XL,SHPXSJ,NOMNELIX,.FALSE.) 00000710NGT=XSJ*WWG(LL) 00000720
C 00000730C COMPUTE RADIUS FOR AXISYMIETRIC CASE 00000740C 00000750
IF (ITYPE.NE.3) GO TO 302 00000760RR:O.00 0000077000 301 ImINEL 00000780
RR:RR+SHP(3,1)*XL(IZ) 00000790301 CONTINUE 00000800
WGT:WGT*2.00P*RR 00000810302 CONTINUE 00000820C 00000830C COMPUTE COORDINATES, VELOCITIES AND GRADIENTS FOR CONVECTIVE TERM 00000840C 00000850
00 32 IZI,NDM 00000860XX(I):O.DO0 00000870V( )0.0 00000880
00 31 K=z,NEL 00000890XX(I):XX(I)+SHP(3,K)*XL(IK) 00000900V(I):V( I SHP(3,K )*UL(I,K) 00000910
31 CONTINUE 00000920YY(LL,I,N) = XX(I) 0000092500 32 J:l,NOM 00000930OV(Z,J)=O.D0 0000094000 32 K21,NEL 00000950
32 OV(I,J)=DV(IJ)+SHP(J,K)*UL(I,K) 00000960C 00000970C COMPUTE NONLINEAR VISCOSITY CORRECTION 00000980C 00000990
XNLN:=l.0 00001000IF (P4.EQ.1.) 60 TO 325 00001010A1:Z.DOO(DV(1,1JN e,0V(2,2)**23+(DV(1,2)0V(2,1))**2 00001020IF (ITYPE.NE.3) GO TO 320 00001030A1=AI+2.00(V(l)/XX(1))**2 00001040
320 XIL!'RzXNLNR/(1.DO.A1*W((1.0-P4)/2.00)) 00001050325 VISLAM x 0.00 00001070
IF (G.EQ.O.D0) GO TO 9 00001072VISLAM : XNLNR*XMU/G+VISLAH 00001076
9 IF (IS*.EQ.6.OR.ISW.EQ.4.OR.ISW.EQ.7) GO TO 47 00001090"C LOOP OVER COLUMNS, FORMING DB, MT*B, AND (DEL.INU)T)ThN 00001100
DO 46 J21,NEL 00001110C 00001120
CALL BMATRX(B,J,ITYPESHPRR) 00001130
93
CALL VtULDF(DsLLBS.NDoIILLB.D8.61 0000114.0CALL VtfLF(XtT(ITYPE13,8,1LLB,NOM,4,LLBXMTB,1,IER) 000011500O 37 IDEX=:11.t1 00001160DO 37 JOEX=1,NDM 00001170IF (IDEX.EQ.JOEX) XN(IDEX,JOEX)SHP(39J) 00001180
37 IF (ZOEX.NE.JDEX) XN(IDEXJDEX3:O.00 00001190CALL VIULPFFDVXNNNfl,NDt1,HDHH,N l,KADVEC.?Ntl,IER) 00001200
JJatJ-13*tzF.1 00001110C 00001220C LOOP OVER ROWS, FORMING BTW(OB3.(MT53TUHTh. AND NT(DEL.(PJ)T)T*t4 00001230C 00001240
DO 45 1u11 N3L 00001250C 00001260
CALL UtATRXB9IsITYPlvSHPRR) 00001270CALL VJLFflh9B5LL8.IiHND1~MnLL56,BTDB,3,IER) 00001280CALL VtIJLFP(XMTIITTPE,1),B.1,LLBNII,*,LLSXIThT,1,1513 00001290CALL ".tDLF"(XMITBTXIT,1.NMNHl,1,1PEN.3,IER3 00001300CALL WIJLFH( XNAOVEC .NDt1,NHf,NOM,NDt1,NDotCAVECNMTER 3 00001310
Z~a(-13*F.100001320C 00001330C ADD TO ELEMENT STIFFNESS MATRIX S(NST,NST) 0000134.0C 00001350
CALL tXA0OJS(IJJ),NSTBTDB,3,Nt,ND)M,WGT*XNLNR) 00001360CALL MXAOS(IIJJI ,HSTPEN,3,NDM,NOM,L:GT*XLAM) 00001370CALL fXADD(S(IIJJ),HSTCAODVEC,N,"t1.NDM,NODM.W GT*RHOI 00001380
C 00001390C ADD THERMAL STIFFNESS 00001400C 00001410
IF (N1.EQ.1) A2zXK*DOT(SHP(1,I),SHP(1,J)NDI) 00001420If LN1.EQ.1) S(11,NDH,JJNDM):S(11,ND1,JJNt),A2*WGT 00001430
4S CONTIllUE 0000147046 CONTINUJE 00001480
IF (1511.EQ.3) GO TO 65 000014S547 CONTINlUE 00001487
IF (ISW.EQ.4.OR.ISW.EQ.6) 60 TO 60 00001497C 00001507C CALCULATE ESIG(LL,2,N): ELASTIC STRESS AT K *1 ITERATION 00001517c 00001527C SET UP A MATRIX FOR PLANE FLOW 00001537C 00001547
AITER(1,1) =OVC1,1)*2.00 000015S7AITER(2.1) a 0.00 00001567AITERC3,1) =.DV(2,1) 00001577AITER(1,2) = 0.00 00001587AITER(2,2) = OV(2,2)*2.DO 00001597AITER(3,2) =OV(1,2) 00001607AITER(1,3) zOV(1*2)*2.DO 00001617AITER(2,3) zOV(2,1)*2.00 00001627AITER(3,3) = DV(1,1) + DV(2,2) 00001637
C 00001647C COMPUTE VISCOUS STRESSES AT GAUSS POINTS: 516 * U 00001657C 0000166760 316(1) 2XMU*2.DOWOV(1,1)*XNLNR 00001677
316(t) * XflU2.D0OI#V(292)*iKLNR 00001687SIG(3) 2 XtIU*(OV(1,2) + DV(2,13))XNLNR 00001697S16(7) z XLAM*(DV(1,1) + OV(2,2)) 00001707
- IF(ITYPE.NE.3) 60 TO 61 00001717SIG(4) z SIG(3) 00001727316(3) 2 (Vt13/XX(1 3)*XMU*XNLNRW2.00 00091737SIG(7) = SIG(7) 4XLAM(VC1/XX(133 00001742
94
61 IP(IS14.EQ.4.CR.ISW.EQ.6J GO TO 62 00001747C 000017S7C CALCULATE VISCOUS STRESS GRADIENT (DEL(SIGIA)) 00001767C 00001777
CALL ESHAP(SG(LL),TG(LLhoXLESHP,NDtl.NELIX) 00001787C 00001797C FORMI CONVECTION DERIVATIVE OF STRESS% ONLY 20) FLOW 00001807C 00002817
00 21 12 00001827DO 21 J=1,3 00001837
21 OOV(JsI) = 0.00 00001847
00 22 K:1,NEL 00001857OOV( , 1) a OOV(1,1) + 2.00*XMU*XNLNR*ESHP(1,K)EUL(1,KI 00001867
DDV(1.21 ODV(1,2) + 2.DOVXMUWJXNLNR*ESHP(3,K)*UL(l,K) 00001877DO21 OV(.1) + Z.D0*XflUfXN4LNR*ESHPt3,K)IiUL(2,X) 00001887
OV(2.2) =DOV(292) + 2.DO*XM.U*XNLNIRIESHP(2,K)*UL(2,K3 00001897DDV(3,1) = OOV(3,l) + XtIU*XNLHRW(ESHP(1,K)*UL(2.K),ESHP(3,KIUUL( 00C019071 1,K)) 00001917DOV(3,21 = OV(3,2) + XMU*XNLNR*(ESHP(3,K)*UL(2K)ESHP(2,K)*UL( 00001927
1 11K)) 0C193722 CONTINUE 00001947C 00001957C SOLVE ESIG(LLZN): ONLY 20 FLOW 00001967C 00001977
ESI61(LL,2,N) =VISLAM*((AITER(1,13*(SIG(1).ESIGI(LL,1,H)). 000019871 AITER(1,2)*(SIG(2),ESIGZ(LL,1,H)),AITER(1,3)*(SIG(3)4ESIG3( 000019972 LL,1,N))-(V(1)*(ODV(1,1),ELAS1(LL,1,N)+V(23*(OOV(l,2),ELASl( 000020073 LL,2,N)l)) 00002017ESIG2tLL,2,N) =VISLAlWI IAITER(2,1)*(SIG(1),ESIG1( LL.1,N) 1. 000020271 AITER(2,2)*(SIG(2).ESIG2(LL,1.Nfl.AITER(2,3)*(SIG(3)4ESIG3( 000020372 LL,1,N)))-(V(1)*(ODV(2,1),ELAS2(LL.1,H)),V(2)*(ODV(2.2),ELAS2( 000020473 LL,2,N)))) 00002057ESIG3(LL,2,H) = VISLA!1w(AITER(3,1)v(SIG(1),ESIG1(LL#,,)), 00002067
1 AITER(3,2)*(SIG(2),ESIG2CLL,1,NJJAITER(3.3)*tSXG(3).ES1G3( 000020772 LL,1,N)))-(V(1)*(0DV(3,1)4ELAS3(LL,1,N)).V(2)*(DDV(3.Z)4ELAS3( 000020373 LL,2,N)))) 00002097
CC UPDATE BOUNDARY STRESSES BOSIG(NODE,0IRECTION,ELIT. NO.)C
IF (G.EQ.0.00) GO TO 65BOSIG(LLP1,N) =ESIG1(LL,2,NI + ELAS1(LL,lvN)*(XL(1,LL)-1 YY(LL91,N)) + ELAS1(LL,2,N)*(XL(2,LL)-YY(LL,2,N))BOSIO(LL,2,N) =ESIG2(LL*,) + ELASZ(LL,1,N)*(XL(1,LL)-I YY(LLPIPH)) +ELAS2(LL,2,N)*e(XL(2,LL)-YY(LL,2,N))BOSIG(LL,3,N) =ESIG3(LL#,) + ELAS3(LL,1,N)"(XL(1.LL)-1 YY(LL,1,N)) + ELAS3(LL,2,N)*(XL(2,LLJ-YY(LL,2,N)IGO TO 85 00002107
C 00002117C PRINT STRESSES IF ISW=4, OTHERWISE BRANCH TO COMIPUTE 00002127C UNBALANCED FCRCE VECTOR 00002137C 0000214762 IF (ISM.EQ.6) GO TO 66 00002157
XIIAX z DtAX(DABS(XL(1*4R-XL(1.1)IDA8S(XL(1,3J-XL(1,2)) 000021581 DABS(XL(2,4)-XL(2,1)).DABS(XL(2,3)-XL(2,2))) 00002159910(5) = (RHO/(XIU*XNLNR) )*DSORYgV(1)4**2V(2)*2)*X(tAX 00002160
- SIG(61 2 *ISLAM*DSQRT(V(1)**2+V( 2)**2)/XMAX 00002161- CALL FPSIG(XXESIG1(LL,2,N),ESIG2(LL,2,N),ESIG3(LL,2,N),SIG, 00002167
1 ITYPENDF) 00002177GO TO 65 00002187
C 0000:197
95
C LOOP OVER NODES TO COMIPUTE U!?43ALANCEO FCTCE VECTOR: 00102207C P =P1 - BT*'SIG - NT*ELASLL,N4)-RH0~t.T(DEL.(t.'UT)TN 003C:217C 000:27
C C~tIUTE UBALANCED TEMPERATURE VECTOR 0023
66 IF (N1.NE.1) GO TO 76 v,00CZ247Q z HEAT*(S!G(1)*OV(1,1).SIG(2)*0V(2Z)SIG(3)(DV(1,2)0V(21.))) 00002257IF (ITYPE.EQ.3) Q 2Q 4 NEAT*(DV(1.2).DV(2,1))*(SIG(4)-SIG(3)) 0000226200 78 Jz1,2 00002:67DLTEE(J) =0.00 0000227700 78 !1l,NEL 00002287
78 DLTEE(J) xDELTEE(J) 4 SI4P(Jtl)*UL(3,i) 0000229776 00 77 11,sNEL 00002307
II x (1-1)*NF.1 00002317C 00002318C CONVECTION TERM SAME FOR 20 AND AXISYMMETRIC FLOW 00002319C 00002320
P1)zP(!!) - RHO*SHPC3,I)*(Vt1)*DVC1,1),VC23*DVC1,2)2*WGT 00002321P(11+1) z P(II,1)-RHO*SHPC3,I)*CV(1)*DVC2,1),VC2)*DVC2,Z))*WGT 00002322IF (ITYPE.EQ.3) GO TO 79 00002324PCI!) =P(II)-(StiPC1,1)4'SIG()SIGC7)).SHP(2,I)*SIG(3))*WGT 00002327PC11+1) =PCIl+lJ-(SHPC2,I)*CS!GC2),SIGC7)),SHIPC1,I)*SIG(3))WT 00002337GO TO 80 00002342
79 PCI!) =P(II)-(SHP(1,I)*(SIGC1),SIG(7)).SHiP(3,1)*SIG(3) 000023431 +SH.4-ZI)*SIG(4) )*'4GT 00C02344P(11+1) =P(II.1)-(SHP(2,I)*fSXGf2),SIGC7)),SP1Ip)SG(4))I4ST 00002345
80 IF (K2.EQ.3.OR.K2.EQ.4) PCI!) =PCIl)-CSHP(3,I)*CELASI(LL,1,N), 000023471 ELAS3(LL,Z,N))l*WGT 0C002357IF (X2.EQ.3.OR.K2.EQ.4) PCII.1) c P(11+1) - CSHP(3,I)*C 000023671 ELAS2(LL.2,N).ELAS3(LL,1,N)) )*WGT 00002377IF (N1.EQ.1) Al =Q*SHP(3.1) 00002387IF (NI.EQ.1) A2z XK*OOT(SHP(1,IIOLTEE,NOl) 00002397
77 IF (Nl.EQ.1) PCII.HDM) =PCU.+NDM) + A1*WGT - A2*WGT 0000240765 CONTINUE 0000241733 CCNTINUE 000024275 RETURN 00002442
E 14"J00002447SUBROUTINE ELMTO6( 0 t UL , XL t IX t TL , S , P ,NDFpNDMNSTISW)00000010
C COO00020
ELT0 ************ ******w**0000000
C 00000055C AN ELEMENT FOR INTERPOLATING DISPLACEMENT, TEMPERATURE, AND STRESS00000060C FOR VISCOELASTICITY: 20 FLOW, OLOROYD DERIVATIVE 00000070
X.C 00000080IMPLICIT REAL*8( A-H ,O-Z) 00000090REAL*S XMIT(4,6)/8*1.00,2*0.D0,2W1.00,12*0.O0/, 00000100I XM(6,4J/2*1.00,4V0.O0,2*1.D0,4*0.OO,3*1.D0,3*0.00,3*1.DO,3*0.O/ 00000110INTEGER LB(4)/3t3t4o6/ 00000120COr.'0N /CDATA/OHEAO(Z0 3,NUrNP,NUMELNUIMAT,NEN,NEQ,IPR 00000130COMM~ON /ELDATA/OM,N,MA,MOTPIEL,NEL 00000140DIMENHSION 0(30),UL(NDF,1hXL(ND,11),IXC1),TL(1), 000001501 SCNtST,1),PC1),SHP(3,9),SGC9),TGC9),WGC 9), 000001602 SIGC7),EPS(6),BSIGC3),XXC3),BC18),08t6,3),BTDBC3,3), 000001703 BU(6),XMTDC3),XMTBT(3),PENC3,3),OUC3),DLTEE(3), 00000180
- 4 V(2),DV(292)PXN(3,S3,ADVECCZ,2),CADVEC(2,2),00V(3,2),CC32), 00000190- 5 XNTN(3,3).ST(2,3),AOSIGC3,2),CN(3,2)oXNITDBC3,2),XNTCN(32), 00000200
6 XNTSTC2#3)9 CAOSIGC3p2l 00000205
IF (ISW.EQ.l) G0 TO 1 00000220
96
ITYPE 0(30) 00000230L D(28) 00000240RHO 0(27) 00000250XLAM 2 0(261 00000260
= 0(26) 00000270XK 0(24 00000200C9= 0(23) 00000290N1 0(0O) 000003C0HEAT 0(19) 00000310LLB = L8(ITYPE) 00000320K2 = 0(18) 00000330N3 = 0(17) 00000340G = 0(16) 00000350P4 = 0(15) 00000360
C 00000370C BRANCH TO CORRECT ARRAY PROCESSOR 00000380C 00000390
GO TO (1,2,3,3,S,3),ISW 00000400C 00000410C ISW = 1: READ MATERIAL PROPERTIES, DEVELOP 0000020C DIAGONAL-STORAGE 0 MATRIX 00000'430C 00000440
1 CALL DFMTRX(D) 00000O50LINT = 0 00000D460
C 00000470RETURN 00000480
C 000004902 RETURN 00000500
C 00000510C ISM s 3: FORM ELEMENT STIFFNES MATRIX 00000520C 00000530
3 CONTINUE 00000540C 00000550C LOOP OVER GAUSS INTEGRATION POINTS 00000560C COMPUTE UNSYMMETRIC STIFFNESS MATRIX 00000570C 00000580
IF (L**NDM.NE.LINT) CALL PGAUSS (L,LINTSG,TG,WG) 00000590DO 33 LL = 1,LINT 00000600
C 00000610CALL SHAPE (S(LL),TG(LL),XLSNPXSJPNMNELIX,.FALSE.) 00000620WGT = XSJ*WG(LL) 00000630
C 00000640C COMPUTE COORDINTAESVELOCITIESi STRESSESP AND GRADIENTS 00000650C 00000660
O0 32 I=l,ND 00000670XX(I)=0.00 00000680V(I)=0.00 00000693
00 31 K:1,NEL 00000700XX(I) XX(I) + SHP(3,K)*XL(I,K) 00000710V(I) S V(I) + SHP(3,K)*UL(I,K) 00000720
31 CONTI UE 00000730DO 32 J=t,NDM 00000740DV(I,J)=O.DO 0000075000 32 K=I,NEL 00000760
32 OV(I,J) = DV(IJ) + SHP(J,K)*UL(I,K) 00000770C 00000780
-C COMPUTE NONLINEAR VISCOSITY CORRECTION 00000790C 00000800
XNLNR 1.00 00000310IF (P4.EQ.1.) GO TO 325 000000
97
Al 2 20*V(.,V22.2,V(,)Dz,)v 00000830
323s VI$LAfl 0.00 00000350IF (G.EQ.0.00) GO TO 9 0COOOC60VISLAI = X?ILNP*XIIU/G +VISLAI 00000870
9 00 320 1=1,3 00000880SIGMI 2 0.00 0000039000 320 J=1,2 00000900OOV(I.JJ = 0.00 0000091000 320 KZ1,NEL 00000920SIGMI = SI1) + SHP(3,K)*UL(NIOF-3,I,K) 00000930
320 DOV(I.J) = DDV(I,J) + SHP(JK)*UL(NOF-3+1,K) 00000940IF (ISW.EQ.4.CR.IS4.EQ.6) GO TO 47 00000950
C 00000960C LOOP OVER COLUKNS FORMI KIT4B,(DEL.(NUJJT*N, BT, 00000970C DEL(N*SIG1A)'*N Nt 08, AND CN 00000980C 00000990
00 46 J=1,NEL 00001000CALL BIIATRX(B,J,ITYPE ,SHP,RR) 00001010CALL CI'.,TRXtC ,J,SIGS)IP) 00001020CALL V,'ULFF(Xnrc(ITYPE,I,B,1,LLB*Nov-394,LLB,Xnre,1,rERI 0000103000 37 IDEXz1.Z 00001040D0 37 JDEX21,^2 Ocoloso
37 ADVEC(IDEX,JOEX) a OV(IDEXJOEX)*SHP(3,J) 0000106000 41 IDEX=1.3 0000107000 41 .DEX=1.3 00001030IF (IOEX.EQ.JOEX) XN(IDEXPJDEX) SNP(3,J3 00001090
41 IF (IDEX.HE.JOEX) XH(IDEX,jOEXI 0.00 W0001100ST(l,1) SHP(1,J) 00001110BT(2,1) 20.00 000011z0BTC1,2) 2 .00 00001130ST(292) SNiP(ZJ) 00001140BT(1,3) a SHP(Z,J) 00001150BTfZ,3) m SHPC1,J) 00001160DO 39 IDEX=1,3 00001170D0 39 JOEX2I.2 00301180AOSIG(IDEX,JOEX) = SHP(3.J)*DDV(X0EX,JDEX) 00001190
39 CN(IDEX.JDEX) % StHP(3,J)*C(IDEX,JOEX) 00001200CALL Vt'ULOF(DLLB,StNII.LLS,DBp6) coc012.0JJ =(J-1),ANoF .1 00001220
C 00001230C LOOP OVER ROM4S, FORMING (MTBVF*MTB, NT(DEL.(NU)T)T*NNT*BT, 00001240C MT(OEL(N*SIGMA)*N, NT*H, NT*O8. AND NT*ECN 00001M5C 00001260
00 45 !:1,NEL 00001270C 00001Z80
CALL 81ATX(B,IYITYPE,SHPlR) 00001290- CALL VIULFF(XIT(ITYPE,1),B,1,LLB,NDF-3,4,LLBXITBT,1,IER) 00001300
CALL VNULFNi(XMTBT,XfiTB,1.N0MN0M,1,1,PEN,3,IER) 0000131000 38 IDEX=1,2 00001320DO 38 JDEX(:1,Z 00001330
38 CAOVECCIOEX,JOEX) = ADVEC(IDEX,JOEX)*SHP(391) 0000134000 40 IOEX=1,2 00001350DO 40 JOEXZI,3 00001360XRTCHtJOEX,IDEX) =CN(JOEX,IDEX)*SHP(3,I) 00001370
- XtITOB(J0EX,IOEX) =08(JOEX,IOEX)*SHP(3,I) 00001380- XNT8T(IOEX,JDEX) = BT(IOFX,JDEX)*SX4P(3,I) 00001390
40 CAOSIG(JOEX,IOEX) 2AOSIG(JOEX910EX)*SNP(t391) 00001400DO 42 IOEX=1,3 00001410DO 42 JOEX=2,3 00001420
98
42 XNTN(I0EX JOEX) * N(IDEXJOE!(I*SHP(3,I) 01001430Il (1-21)NF .1 000 01440
C *00C1450C ADO TO ELEMIENT STIFFNESS MATRIX S(NST,NST) 00001460C 00001470
CALL MXADD(S(11,JJ)NSTPEN3NDI,DII.MSTXLAI) 00001480CALL IIXADO(S( 11 PJJ) JSTCADVEC.NOH,.MNODNWST*RHO) 00001490CALL HXADO(S(II,JJNDF-2),NST.XNTBT,2.2,3,MGT) 00001500CALL MXAOD(S(1100F-2,JJ),ST.CASIG,3,32.GT*VISLAI) 00001510CALL IXAOOS(11II0F-ZJJ),NSTXNTCN,332.-WGTVISLAI) 00001520CALL MXA03(S(11,NDF-ZJJ),NSTXNTD,332,-lGTWXNLNR) 00001530CALL fXAOISEZI~tJF-,JJNOF-2),NST.XNTh,3,3,3,WT3 00001S40
C 00001550C ADD THERMAL STIFFNESS 00001560C 00001570
IF(N1.EQ.1) A2 z XKWODT(SHP(19I),SHP(,J)NDI 00001580IF(N1.EQ.1)3S(11.NOH,JJ.NDM) a S(IINDM,JJ+NDflJA2*WGT 00001590
45 CONJTINUE 0000160046 CONTINUE 0000161047 CONTINUE 00001620
IF (IS14.EQ.3) GO TO 65 00001630C 00001640C PRINT STRESSES IF ISI44OThERUISE BRANCH TO COMIPUTE 00C016S0C UNBALANCED FORCE VECTOR 00001660C 0C001670
IF (13S4.EQ.6) 60 TO 66 00001680X1IAX = DIAX(A(XL(.4)-XL(1,)),DAS(XL(13)-XL12)), 00001690
1 DAS(XL(24-XL(,1)).DAS(XL(23)-XL(2.2))) 00001700SlOCS) 2 (RHO/I XU*XNLNR) )*DS T( V( 1)**2+VI 2 )**2)*XAX 0000171051816) 2VISLAMIVDSQRT(V(1)*W2,V(2)*23/GIAX 00001720CALL FPSIG(XX,0.DO..00,0.oSIG,rrypEdVFri 000017306O TO 65 00001740
C 00001750C LOOP OVER NODES TO COMPUTE UNBALANCED FORCE VECTORS 00001760C P1 = P1 - NTBT*SIGIA - RNO*'NT(EL.NHU)TlT*N 00001770C P2 = P2 - NT(DEL(SI6MA))*V - RT*SIGrIA + NT*0*LOV + NT*NT'V.I*SI6MA00001780
C 00001790C COMPUTE UNBALANCED TEMPERATURE VECTOR 0000180066 IF (N1.NE.1) GO TO 76 00001810
Q= HEAT(SIG()*DV(1,),SIG(2)uDV(2.2)I 00001820DO 7-8 J=1,2 00001830DLTEE(J) = 0.00 00001840DO 78 I:1,HEL 00001850
78 DLTEE(J) =DLTEE(J) + SNPJ*I)VUL(3,I) 0000186076 DO 77 I=1,NEL 00201870
II =I1"~ 4 1 00001880P(Il) =P(II)-(RHO*(V(1)*DVI1.1).V(2'*DVI1.E)).I0OVt1.13 00001090I 1 *DV( 3,2)) )*SHP(3tl)*LST - SHPE1,I)*SI6E 7)*WST 00001930P111.1) = PEII+1)-(RHO*(V(1)*DV(2,1),VIZ)*DVI2.21)(D0VI2.2) 000019101 +OOV(3,1)1)SHP(3.11*WGT - SMP!21)*31S47)ol*T 00001920IF (N1.EQ.1) Al =QwS!IP(391) 00001430IF ( NI.EQ.1) AZ XK*DOT(SHP(l,l),DLTEE.NII) 00001940IF (N1.EQ.1) P(II+NDM) = P(II0HDM) + A1'UIST - At*WGT 00001950P(II.NDPF-2) C P(Il4N3F-2)-(VISLM~W(DDV(1,1)aV(1).DDV(1,ZW*V(2)) 000019601 + SIGhl) - 2.D0wXMU*XNLNR*DV(I9I) - 2.CO*VISLAM* 00001970
- 2 (SIG(1)*DV(1,1),SIG(3)*DV(1,2))S4P(3.I)*UIGT 00001980- P(II.HDF-1) = P(II.NDF-1)-(VISLAtII(DDV(2,1)#Vh1),DDV(2,2)*V(Z)) 00001990
1 4 SIG(2) - 2.D0*Xt*iU*XMLtR*DV(2921 - .DONVISLA1* 000020002 (SIG(2)*DV(2,2).SIG( 3)*DVI2,1) ))*SHP(391)*t4GT 00002010P(II.NDF) 3P(II+NDF) -(VISLAri*(ODVI 3,1)V(1)DVI 3.2)*V(2)) 00002020
99
I +SIG(3) - XlU*XNLNR*IOV(1,2),OV(2,1)) -V!SLAI* 000020302 (SIG ( 2 *OV(1,2),SIG(3)4DV(1.1).SIG(1)VOV(2,1) 000020403 *SIG(3)*DV(2,2) ))*ISP( 3,I)*''GT 0000:050
77 CONTINUJE 0000205565 CONTINUJE 0000206033 CONTINUE 00002070S RETURN~ 00002080
END 00002090SUBROUTINE ESHAP(5,TTXpESNPNDHNELpIX) 00000010
C 00000015C*U*W#NUWWW~WU*~ **NW*****W*~*I~*~** 00000020
~~ ESHAP N~w~*s*wwN~~ 00000030~ 00000040
C 00000050IMPLICIT REALOS( A-H ,O-Z) 00000080
C SHAPE PtANCION ROUTINE FOR 9 NODE QUADRILATERALS FOR SECOND DER. 00000070C 00000080
DIMENSION4 ESNP(3,1),X(NOMtI),SHP(3,9),IX(1),816(3,3),XS(2,2), 000000901 EBIG(3,3)oEXS(3.2),SX(22)TElP(3I 00000100DATA S/0.500/,T/1.D0/,R/2.D0/ 00000110
C 00000120C FORM 9-NODE QUADRILATERAL SHAPE FUtCTIONS FOR SECOND DERIVATIVE 00000130
ESflP(I,1) xS*(TPE4'2-TT) 00000140ESHP(2,1) = *(SS**2-SS) 00000150ESHP(3,1) =S**Z*(RI.SS-T)*(RWTT-T) 00000160ESHP(1,2) =ESHPC1,l) 00000170ESHP(2*2) 2Sw'SS*w2+SS) 00000180ESHP(392) z S*2*(R*SST)*(R*TT-T) 00000190ESHP(1.3) =Sv(TT**2,TT) 00000200ESHP(2,3) =ESHP(2.2) 00000220ESHF(3,3) =S**2*(R*SS+T)*(R*TT+T) 00000220ESHII.4) = ESHP(2,3) 00000230ESHP(2,4) ESHP(2,13 00000240ESHP(3,4) zS**Z*(R*SS-T)*(R*TT+T) 0C000250ESHP(1,5) c -R*ESHP(1,2) 00000Z60ESHP(2.5) = T-SS*s2 00000270ESHP(3,5) =SS*(T-R*TT) 00000280ES4P(1,61 T-TT**2 00000290ESHP(2v63 -R*ESHPC2,2J 00000300ESHP(3,6) 2-TT*(R*SS.T) 00000310ESHP(2,7) =-R*ESNP(1,4) 00000320ESHPCZ.7) =ESHIP(2,5) 00000330ESHiP(3,7) =-SS*(R*TT+T) 00000340ESHP(1,8) ESNP(l.6) 000003S0ESNP(2,8) -R*ESHP(2,I) 00000360ESHiP(3,8) TTWA(T-R*SS) 00000370ESHP(1,9) z -Ri*ESHP(1,61
- ESIP(2,91 -R*ESNPE2,5)ESHP(3,91 RMN2*SS*TT
C 00000380C CONSTRUCT BIG MATRIX AND ITS INVERSE 00000390C 00000400
CALL SHAPE(SSTTX,SHP,XSJ,NWMNELIX,.TRUE. 2DO 130 ZIHON 0000041000 130 JC1,2 -00000420
- XSII,JI z 0.00 00000430- DO 130 Kz1,NEL 00000440
130 XS(I,J) XS(IJ) + XIIK)*SHP(JtK) 000104SOBIG(1,13 z XS(1,1)**2 00000460B10(2,1) = XS(2,1)**Z 00000470
100
S10(3,1) XS(1,1)*XS(2,1) 004SSIGtl,2) XI,)*200000490BIG12,2) 2XS(2,2)** coOcCscoB10(3,21 XS(1,2)*XS(292J 00000510610(113) 22.00*Xs(1,11*XS(1,I) 00000520S10(2,3) 2 .00*XS(2,1)*XS(2o2) 0000053061600) 2XS(1,1)*X3(2,2) 4 XStl*x51291) 00000540
C CALCULATE DETERINANT OF BIG 00000550CET S 1(,)(I(.)8033-!t,)5023,BG21* 00000560
2-SIG(Zo2)*8Z6(1.3)) 000003"oC 00000590C FORM INVERSE 000006CO
ES10(1lll (BIG(2,2)*B10(393)-BIG(32)*51G(2,3)/DET 00000610EBIG(Z,1) 2-(BIG(1.2)*'5IG(3.3)-8IG(3.2).'BIG(1,3))/VET 00000620EBIG(3,11 (BiG(1,23N816(2,33-BrG(2,t)5ZG(1,31?/OE? 00000b30EtIG(I.2) -10 .)0G33-I3,)D62I/CT00000640EBIG(292) 2(BIG(1,1)*BIG(3,3)-8IG(3,1)*51S(2.3)/DET 00000650EOmc(3,21 ODII(,)3023-IG21~G13)DT0CO0660EtIG(1,31 EG21803Z-I(,)I(.)/E 00000670
Et3IG(3*3) 2 BG11*I(.)BGZ14G12)~T00000690C 00000 700C FOR" SECOND DERIVATIVE MATRIX 00000710C 00000720
00 131 1=1,2 00000730DO 131 J=1,3 00000740EX(S(.J,I) --0.00 00000750DO 131 K21,NEL 00000760
131 EXS(J,I) = EXS(JoI) + X(I,KW3ESHP(J#K) 00000770C FORMI JACOSrAN KATRrX INVERSE 00000780C 00000790
SX(1,1) 2XS(2.2)/XSJ 00000800SX(1^2) 2 YS(i.1)/XSJ 00000810SX(1,2) = -XS(1,2)/XSJ 000008zoSX((211 = -XS(ZdJI/XSJ 00000830
C 00000840C FORM GLOBAL SECOND DERIVATIVES 00000850C 00000860
00 132 I21,NEL 00000870TEKP(11 = ES(P(,I) 00000880ITEMP(23 a ESHP(2,11 00000890TEMP(31 a ESHP(3,I) 00000900DO 133 Jm1.3 00000910ESHW(J,I) a0.00 0000092000 134 K=1.3 00000930ESHP(Ji) ESHP(JoI) 4 ESIG(JtK)*(TElP(K)- (EXS(Kpl)*ISXtl,1)* 00000940I 1 gPE1Il).5X(1,2)*SHP(2,1)13-(EXS(K,21I$q5X(t1)*HP(1,l).3X(2,2)* 000009S02 SNII))0000960
134 CONTINUE 40000970
133 CONTINUE 000001*80132 CONTINUE 00000990
RETURN 00001000E14D 000@1410SUBROUTINE PFORIII UL #XL,9TL,9 LD ,PsS, It v0,10, 00000010I X XX , F 9TJIAG p 0vA CNDF* 0000020
- 2 NDMNENlHST.ISWUUUAPLSPLCFPDPL) 00000030C COMPUTE ELEMENT ARRAYS AND ASSEMBLE GLOBAL ARRAYS 00000040C 40000050
101
C~~**w***N ~PFORM * NNW ~ 7N**********0000030
C 00000090IMPLICIT REAL*8( A-N .O-Z) 00000100
LOGICAL AFL,BFLPCFLOFL 00000110COION /COATA/ 0,HEADOCO,HttPHWtELIIDIIATNE4.tEQ,IPR 00000120CONMON /ELOATA/ 0?1.NtMAMCTpIELsNEL 00000130COMM1ON /PRLOO/ PROP 00000140COM1.0 /FVISC/ KZ 00000145
I CONVOIN /TAYLR/ ESIG(4,2.503.ESIG2t4,2,503,ES1G3(4,2,501, 000001501 YY(4,2,503.ELAS1(4,2,50),ELAS2(4,2,503,ELAS3(4,2,S03, 000001602 BOSIG(4,2,503
DIMENSION XL(N0M,13,L(NF13P13.(NT,),IE(13,Df30,13,ID(1MF1000017013,X(NDM,),IX(NEN1,),F(N0F,13,JOIAG(13,B(1)A(3,C()UL(NIWF1D 000001802 9TL(1),T(1),U(13,UD(NOF.13 00000190
C 000 00200
IFL(K2.LE.2.OR.K2.EQ.3).OR.(ISW.LE.4).OR.CN0F.GE.4)) 60 TO 102 00000210
C SET ITERATION PARAMETERS FOR FLUID VISCOELASTICITY 00000:30
NSTEP 0 000002501l .E.1COO06
C BEGIN VISCOELASTIC ITERATION: LOOP ON ELEMENTS 00000290C 000003005 IEL 20 00000310
* 00 101 N z1,NUtIEL 00000320C 00000010'C CALCULATE ELAS WITHIN ELEMENTS USING CENTRAL DIFFERENCES; 00000020C THESE WILL BE USED FOR BOULD0ARY ELEMENTS 00000030C OO00O045C GAUSS POINT 1 00000050C 00000C60
AA c YY4ZN)-X(2,IXl1,N)) 0000007089 YY(2,2,N3-X(2olX(IN)) ocooCOoocc x YY(2,1,N3-X(1,IX(1,N)3 00000090DO YY(4,1,N)-X(1.IX(1,N13 00000100ELAS1(1,1,N) ((ESIG1(2,1,N)-BOSIG(1,1,N33*AA-(ESIG1(4,1,N3 00000110I -BOSIG(1,1,N) )*BB)/(CC*AA-BB*003 00000120ELAS1(1,2,N) =((ESIGI(4,1,N3-BOSIG(1,1,N)3*CC-(ESIG1(2,1,N) 000001301 -EOSIG( 1,1,N) )*00 3/(CC*AA-88N003 00000140ELAS2(1,1,N) - ((ESIG2(2,1,N3-BOSIG(1,2,N)*AA-(ESIG2C4*1,N 000001501 -BOSIG(l,2,N3 3W6B)/(CC*AA-BB*O00 00000160ELASZ(1,2,t4)z ((ESIG2(4,1,N)-BOSIG(1,ZN)WCC-(ES162(2,1,N) 00000170
1 -BOSIG(1,2,N) )*0 3/(CCNAA-BBWOO) 00000180ELAS3(1,1,N) = (EsrG3(2,1,N3-BOSIG(1,3N)WAA-(ESIG3(4,1,NI 00000190
1 -BO3IG(1,3,N))*P.B)/(CC*AA-BOIIDO) 00000200- LAS3(1,29N) 2 CES341,N)-BOSIG(1,3,N))*CC-(E5163(2,1,N 00000210
1 -BOSIG(1,3,N)wOO)/(CCAA-BB*00) 00000220C 00000230C GAUSS POINT 4 00000240C 00000250
AA 2X(291X(4#N33-YY(1,ZN) 00000260SBB YY(3.2,N)-X(2,IX(4,N)) 00000270CC 2 YY(3t1,N)-X(1,IX(4,N)) 0000028000 = X(1,IX(4*N33-YYC1,1,Nl 00000290
-: ELASI(4p1,N) m ((ESIG1(3,1,N)-UOSIG(4,1,N)3'AA-(SOSIG(4.1,N) 000003001 -ESIG1(1,1,N)3WBB,, (CC*AA-B8W003 00000310ELAS(4,2,Nl (SOSIGC4,1,N3-ESIG1I1,1,N))CC-!SGlt31.t4) 000003201 -SOSIG(4,1,N3 3*00 /(CCWAA-B*0D 3 00000330
102
ELAS2(4,1,N) ((ESIG2(3,1,N3-BOSIG(4,2.N3)IAA-t50SI6I4,Z,N) 000003401 -ESIGZ( 1.1,N3))*r 3/( CC'AA-83.OD I 000003s0ELA!!^t4,2,N) ((BOSIG(4,2.tI)-ES!GZ(1,1,HIiW)CC-(ESIG(31,t3 00C0060o
1 -EC3IG(492,N) 3D)/CC*AA-B3tOO) 30Co30OELAS3(4,1,N) x ((ESIG3(3.1,N3-BOSIG(4,3.N33*AA-(BOSI(4,3N 0000038001 -ESIG3(1,1,N) )*EB)/(CC*AA-8S3'OO) 00000390ELAS3(4p2,NI 2 ((BOSIG(4,3,Ni-ES103(1,1,N))*CC-(ESZG3(3,1,N) 000004001 -8OSIG(4,3,N3 3*003/(CC*AA-BB*OO) 00000410
C 00000'420C GAUSS POINT 3 00000430C 00000440
AA z Xt2ZX(3,N))-YY(2oEN3 00000450BB z X(2,IX(3,N))-YY(4,Z,N) 00000460CC c X(1,IX(3.N))-YY(4,1,N) 0000047000D X(1.IX(3,N))-YY(2,1,N) 00000480ELASI(3,1.N3 = (BOSIG(3,1,N)-ESIGZ(491,NII*AA-(5OSZG(3,1,NI 000004901 -ESIGiC 2,1,N) 3*ES3/(CC*AA-B3NOD 3 00000500ELASI(3,2,N) z fEBOSIG(3,1,N)-ESIGI(2,1,N))*CC-(BOSIG(3,1,N) 000051
1 -ESIGI(4,1,N) *3) /(CCNAA-BB*0O) 00000520ELAS2(3,1,N) m C(BOSIG(3,2,N345S162(4,1,N)J*AA-(BSZG(3.2,NI 000005301 -ESIG2 2.1.N3 )vrB)(Cc*AA-55*OD3 00000540ELAS213.2,N) ((EOSIG(3,2,N3-ESIG2(2,1,N))*CC-(BOSIGC32.N 000005501 -ESIGZ(4,1,N3 3*0D3/(CC*AA-BD'iDD) 0o000o:-ELAS3(3,1,NI 2 CCBOSIG(3,3,NI-ES163(491,N33*AA-(BOSIG(393,N) 000005701 -ESIG3C 2,1,N3 3)fBB/(CC*AA-B05O0 3 00000530ELAS3(3,2,N) z EBOSIG(3,3,N3-ESIG3CZ,1,N)3 CC-IBOSIG(3*3N) 000005901 -ESIG3(4,1,N) )*OO /(CC*AA-85*00) 00000600
C 00000610C GAUSS POINT 2 00000620C 00000630
AA 2YY(3v2,N)-X(Z.IX(2pN)) 0000064065 = X(2,IXCZ,N))-YY(1,Z,N) 00000650CC = XC1,IX(2,N33-YY(1,1.N3 00000660DO0 YY(3,1,N)-XC1,IX(2,N33 000C0670ELAS1(2,1.NJ (CBOSIG(2,1,NJ-ESIGl(1,1,N))*AA-(ESIGlf3,1,N) 000006801 -BOSIGI 2.1,N) 3N5 /(C CAA-BBwOO 3 00000690ELAS1(2.2vN) ((ESIG1C3,1,N3-BOSIGCZ,1,N3)NCC-(BOSIGC2,1,N 00000700
1 -ESIGIC 1,1,N3 3q003/(CCAA-B!3wOD3 00000710ELAS2(2,1,N) 2 ((BOSIG(2,2,N)-ESIG2C1,1,N))*AA-(ESIG2C3,1,N3 000007Z0
1 -BOSIG(292,N3 3NBB3/(CC*AA-Bf3*OO) 00000730ELAS2(2,2*N) 2 (CESIG2C3,1,N3-BOSIG(2,2,N))*CC-(BOSIG(2,2,N) 00000740I -ESXG2(2,1,N))*0D 3/(CC*AA-B8NDD) 00000750ELAS3(2.1,M) z E(COS!G(2,3,N3-ESIG3(1,1,H33NAA-(ESIG3C31,NI 000007601 -BCSIG(2,3,N3 3W883/(CC*AA-BS*DO) 00000770ELAS3C2,2,N) z ((ES1G3C3,1,N)-BOSIGC2,3,N33*CC-(BOSI6C2,3NI 00000780
1 -ESIG3C 1,1 ,N) I*DO /C CC*AA-BBVDD 3 00000790C 00000800
-C REPLACE ELAS FOR INTERIOR ELEMENTS 00000810C 00000820
00 91 IDEX x 1,NUCIEL 0000033000 92 JOEX c 1,NUIIEL 00000340
C 00000350C GAUSS POINT 1 00000360C 00000370
!F((IXC1,N3.NE.IX(4,IOEX)).OR.CIXCZN3.N!.IX(3,ZDEX)3 360 TO 10 000003800- IF((IXC1,N3.NE.IX(2,JDEX)).OR.(IX(4,H).NE.IX(3,JDEX)3)GO TO 10 0000390
A A x YY(492,N)-YY(4#2tI0EX) 00000400B z YYI2#2*N)-yy(29ZJDEX3 00000410CC =YYC2,1,N3-YY(2l1,JOEX3 0000042000 Z YYC4,IPN)-YYC4,1*IDEX) 00000430
103
.- -t.
1 ESIG(,I0!X))3ESI/(CC1,NA-ESt I219-X)*A-EIO),1 00000440
1 -ESIGI(2,1.oIEX) )'0O)/(CCWAhA-C'DD00000OO470ELASM1,11 -- ((ESIGI(2,1,N3-ESIG1(2,1,JOEX))*CA-(ESIGI(4,1,N) 00000480
1 -ESIG12,1.JOEX) )w28)/(CC*AA-E2IDD) 00000490ELAS211#1.N3 - I(!51G2,1,N)-ESIG6241,IOEX))*CC-(ES1GZIZ.1,N) 0000050
1 -ESIG2(411,JOEX) 1w0O)/(CC*AA-E8"OO) 00000410ELAS3L1,1.HJ % fESIG3(4,1,N)-ESZG3(4,IPJOEX))*AA-(ESIG212,1.N) 00000520
1 -E51G3(4,1,IOEX3 305)/(CC'eAA-!B"'OO) 00000530ELA33(2,29N) 2 ((ESIG3t4,1,N)-ESIG3(4,1pI0EX))WCC-IES1G3(2,1,N) 000005401 -ESIG3t2,1,JOEX3)NOO )/(CC*AA-BBOD) 00000550
C 00000560C GAUSS POINT 4 00000570C 0000053010 IF((IX(1,N3.NE.IXfUIOEX)).OR.(!X('4,N).HE.ZX(3,IDEX)3360 TO 20 00000590
IF((rx(3,NL.NE.IX(2,J0EX)).OR.(IX(4,N).NE.ZX~1,JO2X)))GO TO 20 00000600AA YY(ls2,JDEX)-YY(1,2,N) 0000061055 = YY(3,2,NJ-YY(3,Z,IDEXJ 00000620CC z YY(3,1,N)-YY(3,1,IOEX) 00000630DO = YY(1~1.JDEX)-YYt1s1*N3 00000640ELASI(4,1,N3) ( (ESIGI(3,1,N)-ESIG1(3,1,IDEX))*AA-(ESIG1(1,1,JDEX)000006501 -ESI1 ( 1.1 N) )*'t /( CCI-AA-Bl' 0O 00000660ELASI(4,2,N) = ((ESIGIt1,lJD)EX)-ESIG3(1,1,N)3*CC-(ESIG1(3,1,N) 000C06701 -ESIG1(3,1,IOEX) )*00)/(CC*AA-88*00I 00000680ELAS2(4,1,N) (IESIG2(391,N3-ESIG^ (3,1,IOEX))*AA-(ESIG2(1,l.JOEX)000006901 -ESIG2(1,1,N3)wB83/(CC*AA-8BBY 00) 000007C0ELASZ(4,N) a ((ESIG2(1,1.J0EX)-ESIG2(1,1,Nl'*CC-(ESIG(3,1,I 000007101 -ESIGZ(3,1,IOEX) OO0)/(CC*AA-D8*00) 00000720ELAS3(4,1,H) =((ESIG3(3,1,N)-ESIG3(3,1,IOEXr)*AA-(ESIG3(1,1,JOEX)000007301 -ESIG3(1,1,N) )*B)/(CC*AA-BB*'0O) 00000740ELAS3(4,2,N) 2 ((ESIG3(1,1,JOEX)-ESZG3C1,1,N))*CC-(ESIG3(3,1,N) 000007501 -ESIG3( 3,1,IOEX) )*DD,/(CC*AA-EB5*00) 00000760
C 00000770C GAUSS POINT 3 00000730C 0000079020 IF((IX(3,N).NE.IX(2,!OEX)).OR.(IX(4 N3.NE.ZX(lolEX)))GO TO 30 00000800
IFU(IX2,N).NE.IX(1,JDEX)).CR.tIX(2oN).NE.IX(4,JDEX)))GO TO 30 00000810AA 2 YY(2FZ,IOEX)-YY(Z2*N) 0000032055 = YY(4,2,JDEX)-YY(4,2oN) 00000830CC = YY(4,1.J0EX)-YY(491,N) 0000084000 2 YYE2.19IDEX)-YY(Zt1,N) 00000850ELAS1(391PN) --((ES1G1(4,1,JOEX)-ESZG1(4,1,N))*AA-(ESIG1EZ,lIOEX)000008601 -ESIG1(2,1,N))ff88)/(CC*AA-B8*00) .00000870ELAS1(3,2pN) --(!SIG1(2,1,10!X)-ESI1(2,1,N))*CC-(ESIGI(4,1,JOEX)000008301 -ESIGI(4,1,N))*t0O)/(CC*AA-0B*OO) 00000390ELASZ(3,1,N) -- ((ESIG2(4,1,JO!X)-ES162(4,1,N))*AA-(ESIG2(2,1.IO!X)00C009001 -ESIZ(2,1,N3 i*eBI/(CC4IAA-SS*00? 00000910E1AS2(3,2,N) m((ESIG2C2,IoIEX)-ESIG2(2,1,N3)*CC-(ESXG2(4,1,JDEX)00000920
1 -ESIG2(4,1,NI JNOO /(CC*AA-8B*00) 00000930ELAS3(39,N) -- ((ESIG3(4,1,JDEX)-ESIG3(4,1,N))*AA-(ES1G3(2,1,olEX)000009401 -ESI63(2,1,N) )*BB)/(CC*AA-BB6O00 00000950ELAS3(3,2,N) -- (ESIG3(2,1,IOEX)-E51G3C2,1,N))*CC-(ESIG3(4,1,JOEX)000009601 -ESIG3(4,1,N) )*0D)/(CC*AA-88E00 3 00000970
C 00000980-C GAUSS POINT 2 00000990-C 0000100030 IF((IX(2,N).NE.IX(1,ZDEX)).OR.(IX(3,NI.NE.IX(4,orEXJ3)GO TO 92 00001010
IF((IX(1,N).NE.IX(4,JOEX) ).OR.(IX(2,N).NE.IX(3,JDEX)))GO TO 92 00001020AA aYY(3,2,14)-YY(3*2,JOEX) 00001030
104
es=YY(1,2,IOEX)-YY(1,2,N) 00001040CC = YY (1,1,IOEX)-YY(I,1,N) 00001030DO0 YY(3,1,N)-YYt3,1,JDEX) 00001060ELASI(2,1.N3 a ((ESIG1(1,1,IOEX3-ES!G1(l,1,N)I'AA-(EIG1(3,1,N3 00001070
1 -ESIGlE 3,1,JOEX) )e1B)/(CC*AA-eB*0O3 00001080ELAS1(Z22N) ((ESIC.(3,1N3-ESIGI(3s1.J0EX)CC-(ESIG1C1,1,!0EX)00001090
1 -ESIG1( 1,1 ,N) )*0O)/( CC'*AA-B8B00 3 00001100ELAS2(Z.1*N) ((ESIG2(1,1,IDEX)-ESIG2(1,1,N))*AA-(ES!G2(3,1,N3 00001110
1 -ESIC-2(3.lJOEX) 3*BB)/(CCwAA-0'B*OD) 00001120ELASZ229N3 ((ES1GZ(3,1,N)-ESIG2(3,lJDEX3))CC-(ES1G2(lo,,3EX)00001130
1 -ESIG2(1,1,N3))W0O)/(CC*AA-B0'*ODI 00001140ELAS3(Z,1,N) z (ESIG3(1,1,IOEX)-ESIG3(1,1,NI)*AA-(E5163(3,1.N) 00001150
1 -ESIG3( 3,1.JOEXI))*BB3/CC*AA-B8*0O) 00001160ELAS3(2,2,N) =((ESIG3(3,1,N)-ESIG,3(3,1PJDEX))*CC-(ESG3(,1,ZDEX)00001170
1 -ES1G3(1,1,N) )*O0)/(CC*AA-55*00) 0000118092 CONTINUE 0000126091 CONTINUE 00001270C SET UP LOCAL ARRAYS FOR CALCULATING ESIG(LL92,N) 00001280
00 55 I:1,NEt4 0000129011 = IXLI,N) 00001300IF (II.NE.0) GO TO 55 00001310TLI) = 0. 000013200O 53 J=1,NO?1 00C01330
53 XL(J,I) = 0. 0000134000 54 J21,NOII 00001350UL(J,I) = 0. 00001360UL(JI.NEN) = 0. 00001370
54 LO(JI3 0 00001330GO TO 58 00001390
55 110 = II*MOF-NOF 00001400NEL C 1 00001410TLI) x T(II) 0000142000 56 J=1,NDII 00001430
56 XL(J,4) = X(J.11) 0000144000 57 J=1,?NOF 00001450K a IASC!(J,II)) 00001460UL(J.1) F(J,II)*PROP 00001470UL(J,I+NENI x UD(J,II) 00001480IF (K.GT.0) UL(J,I) = MC) 00001490IF (OFL) K = rID + J 00001500
57 LO(J,Z) a K 0000151058 CONTINUE 00001520C FORM1 ELEMENT ARRAY 00001530
MA = IX(NEN1,N) 00001540NIF (ZECCIA).NE.IEL) HCT = 0 00001550
IEL = IE(IIA) 00001560CALL ELMIB(0(1,11A),ULXLIX(1,N),TLSPNDFHDINST,7) 00001570
101 CCN'TIN'UE 00001580YM-AX =1AX(ABS(ESIG1(1,Z,1)-ESIG1(1,1,13),0A8S(ESIG2(1,Z,1) 00001590
1 -ESIGZ(1,1,1)),OABS(ESIG3(1,2,1)-ESIG3(1.1,1))) 0000160000 93 I:1,NUIEL 0000161000 93 J=1,4 00001620XtIAX :OMAX1(DASS(ESIG1(Jt,,!3ESIGItJ.1,I),ABStESI62(J,2,I) 000016301 -ESIG2(J,1,!)),OABS(ESIG3(J,2d1)-ES103(J,1,Z))) 00001631
93 If CXIAX.GT.YMAX) YIIAX=XMAX 00001632- IF'fYtiAX.Lf.TlDL1) GO TO 102 00001633- NST!P z NSTEP + 1 00001634
00 90 K1,tNIIEL00 90 J=1#4ESIG1(J,1,K) ESIGI(J,2,K)
105
ES!G0J1,K) ESIGZIJ,2,K)90 ESIG3(J,l.Kl ESIG3(J.2,K)
IF INSTEP.GE.10) GO TO 102 00001635IF (NSTEP.EQ.1.OR.N.STEP.EQ.3.OR.NSTEP.EQ.5. 000016361 OR.NSTEP.EQ.9) GO TO 94 00001637Go TO 5 00001638
94 WRITE (6,1000) O,HEA09TItIE,NSTEP 00001639WRITE (6,1010) 0000164000 95 l:1,WrtEL 00001641
95 WITE (691020) 1,((YY(J,1,1),YY(J,2,I),ESIG1(J,2,I) 000016421 ,E5IG2(J,2,1).ESIG3(J.2,I)),JI'1,4) 00001643GO TO 5 00001644
C LOOP ON ELEMIENTS: ELASTIC ITERATION COMPLETE 00001649102 CONTINUE 000016S0
IlL z 0 0000166000 110 N 21,NUIEL 00001670
C SET UP LOCAL ARRAYS 00001680DO 108 I l,HEN 0000169011I= X(I.N) 00001700IF (II.NE.0) 6O TO 105 00001710TL(I) =0. 00001720DO 103 J=1,NOI 00001730
103 XL(J,I) 0. 0000174000 104 J =1,NVF 00001750UL(J,I) =0. 00001760UL(J,I+NEN) 0. 00001770
104 LO(J,I.) = 0 00001780GO TO 108 00001790
105 110 II*HOF -NOF 00001600NIL zI 00001810TL(I) =T(II) 0000181100 106 J=2,NDHt 00001812
106 XLIJ91) X(J,11) 00001813DO 107 J=1,NOF 00001814K mIABS(ID(J,11)) 00001815UL(J,I) 2F(J,II)*PROP 00001816UL(J,I+NEN) =UO(J,II) 00001017IF (K.GT.0) UL(J,I) =U(K) 00001818IF (DPI) K = ID + J 00001819
107 LD(J,I) =K 00001820108 CONTINUE 00001821C FORMI ELERMENT ARRAY 00001822
MIA zIX(N!N1,N) 00001823IF(IE(tIA).NE.IEL) tICT 0 00001824IEL zIEttIA) 00001825CALL ELtILIS(O( 1,tiA) ULXL,IX( 1,N) ,TLSPNOFNDI1,NST,ISW) 00001826
C ADO TO TOTAL ARRAY 00001827- IF(AFL.OR.BFL.CR.CFL) CALL ADOSTF(AB,CSPJDIAS,LONSTNEL*NDF, 00001028
1 AFL,8FL,CFL) 00001829110 CONTINUE 000018301000 FORM'AT(Als20A4,//5X,'ELASTIC FLUID STRESSES AT GAUSS POINTS', 00001831
L 1 SX, TIIE ,G13.5,//1X, VISCOELASTIC ITERATION NUtIBER:' ,I4//) 00001832F . 1010 FORt)AT(IX, ELMT' ,11X, '-CCORD' ,6X, 2-COORO' .20Xs 00001833
1 *ETAU-XX' ,7X, ETAU-YY ,7X, ETAU-XY'//) 000018341020 FC'4ttAT(IS,/4(IOX,2G13.4,13X,3Gl3.4/)//) 00001835
RETURN 00001849r - Ell 00001850SUBROUTINE CIIATRXtCvJ ,SIGSNP) 00000010
C 00000020W*NN*~WW***W*WN*W**W*W* 0000030
106
... CIATRX ..... .........N 00000040*W*C*** 00000050
C 00000060IMPLICIT REALVSIA-",O-Z) 0000070DIMENSION C(13,SIG(7)tSHP(3,9) 0000080
C 0000090CALL PZERO(Co6l 00 000
C 0:000110C ONLY 20 FLOW TREATED HERE 00000120
C CUl) z2.0(SIG(lW*SHP(1,J),SIG(3)*SHP(2,J)) 00000::140:
C ' = 0.00 00000150C(3) 2SIG(2)*S14P(2pJ)4SIG(3)*SHP(IPJ) 0 0006C(41 0.00 000170C(S) =2.0*(SIG(2)*SHP(2oJ),SIG(3)*SHP(lJ)) 00000180C(6) SIG(13*SHP(1,JSIG(3)*SIP(2,J)l 00 000190RETURN 00000200ENDO
0 0000210SUBROUTINE FPSIG (XX.ESIG1,ESIG2,ESIG3,SIG,ITYPE,NOF) 00000010
C 00000020C~* W* ~ *~ FPSIG * * V * * * 00000030
*NW* **** *~*~ 00000040C 00000050
IMPLICIT REAL*S( A-N ,O-Z) 00000060DIMENSION XX(1) ,SIG( 1) 00000070COM;ON /C0ATA/ O,HEAO( 20),NUMNP,NUIEL,NUtiAT,NEN,NEQIPR 00000080COMW-ON /ELDATA/ Dtl,NMAiMOTIELNEL 00000090COMMON /TOATA/ TIIE9OT,C1,C2,C3tC4,CS 00000100COMMlON /FVISC/ K2 00000110
C 00000120GO TO (51,52,53,54), ITYPE 00 000130
C 00000140C PLANE FLOW 00000150C 0000016051 MOTmMOT-1 00000170
IF (KZ.LE.2) GO TO 509 00000240A =SIGt1) + ESIGI 00000250B SIG(2) + ESIG2 00000260C =SIG(3) + ESIG3 00000270
509 IF (rOT.GT.0) GO TO 510 00000180IF (NOF.LT.4) WRITE (6,5000) ONEADTIME 00000190
5000 FORMAT (AI920A4,//5XP*FLUID VISCOUS STRESSES AT GAUSS.POINTS:', 000002001 SXTINE'PG13.5, 000002102 //lX,'ELIT tATL',6X,'l-COORO',6X,'2-COORO',SXv 000002203 'FRESSURE',7X,'TAU-XX',7X,'TAU-YY',7X,'TAU-XY'/) 00000230IF (NDF.LT.4.ANO.K2.GE.3) WRITE (695010) 00000280
5010 FORMAT (//5X, 'TOTAL VISCOUS AND ELASTIC STRESSES ATGAUSS POINTS: 00000290I,/) 00000300IF (KOF.GE.4) WRITE (6,5020) O,I4EAO,TIIE 00000301
5020 FORMAT (A1,20M4,//5XP'TOTAL VISCOUS AND ELASTIC STRESSES AT 000003021 GAUSS POINTS:',SX,'TIME',G13.5, 000003052 //1X,'ELIT IATL',6X,'1-COORD',6X,'2-COORD',SX, 000003063 *PRESSURE',7X,'TAU-XX',7X,'TAU-YY',7X,'TAU-XY'/) 00000307IF (NOF.GE.4) GO TO 509 00000308IF (K2.GE.3) MOT:19 00000310IF (K2.GE.3) GO TO S0S 00000320
- MOT z S 00000330508 CONTINUE 00000350510 IF (NOF.LT.4) WRITE (6,5001)N,?1A.XX(),X(X(2),SIG(7),(SIG(I),1=1,3)00000360
IF (NOF.GE.4) WRITE (6,5009)N,MA,XX(1),XXC(2) 00000363
107
._ _ .6
5009 FOPt1AT (215,2G13.4) 00000366IF (KZ.GE.3) IVRITE (6,5011) SIG(5)qS!G(6),AvBC 00000370
5001 FC!,.AT (^,15,6G13.4) 000003305011 FCR!IAT (1X,'RE a',Gl3.4,OUS ,tG13.4913X93G13.4) 00000390
RETURN 00000400C 0000041052 RETURN 00000420C 00000430C AXSY1METRIC FLOM 00000440C 0000045053 IOT=MOT-1 00000460
IF (tIOT.GT.0) 60 TO 530 00000470WRITE (6,5002) OtHEA0,Tlt! 00000480
5002 FORMAT (A1,20A4*//5Xo.FLUID STRESSES AT GAUSS POINTS:%s 000004901 SX*TfIE,6G13.5, 000005002 //1Xt*ELMT H4ATLt6X,1l-COORD*,6X2-COOR',5X, 000005103 'PRESSUE,97X,'TAU-RR,97X,'TAU-ZZ',7X,'TAU-TTp7X.'TAU-RZ'/) 0000052011OT a 50 00000530
C 000005.40530 WRITE (6,5003) NIAXX(1),XX(2),SIG(7),(SZG(I),12'1,4) 000005505003 FORMAT 1215,7623.4) 00000560
RETURN 00000570C 0000058054 RETURN 00000590
END 00000600
108
APPENDIX 4
Input Data Set Listings
1. Run 1 - Linear Cross Channel Flow
18-9 Node Elements
2. Run 3 - Linear Cross Channel Flow
72-8 Node Elements
3. Run 4 - Convection (Re = 0.4) Cross Channel Flow
18-9 Node Elements
4. Run 6 - Viscoelastic (Ws = 0.02) Cross Channel Flow
18-9 Node Elements
5. Run 13 - Viscoelastic (Ws = 0.001) Entry Flow
24-9 Node Elements
6. Run 20 - Linear Entry Flow Fully Developed Boundary
Conditions 24-9 Node Elements
(Note: Run 20 is not listed in Table 1)
109
* Z-
UI
BRC4066 (FOREGROUND): OUTPUT FROM TSO XPRINT Input Dataset Run No.
AT 13:42:06 ON 12/05/00 - !RC4066.TEST.OATA
FEAP CROSS-CHAHNEL FLOW - NEWTONIAN (TEST 7) 0000001091 18 . 2 2 9 0 00000020
COOR 000000301 7 000 000 00000040
85 0 200 000 000000502 7 000.166666700 0000006086 0 200.166666700 000000703 7 000.333333300 0000008087 0 200.333333300 000000904 7 000 .SDO 00000100
88 0 200 .300 000001105 7 000.666666700 0000012089 0 200.666666700 000001306 7 000.833333300 00000140
90 0 200.833333300 000001507 7 000 100 00000160
91 0 200 100 0000017000000180
ELEM 000001901 1 1 1S 17 3 8 16 10 2 9 14 000002C07 1 3 17 19 5 10 18 12 4 11 14 0000021013 1 5 19 21 7 12 20 14 6 13 14 00000220
0.0000230MATE 00000240
1 S NINE-NODE LAGRAGZAN PENALTY ELEMENT 000002501 0 2 1 1.0 000002602 .1000+009 .7900.003 .0000 00000270
00000275BOUN 00000Z80
1 7 -1 -1 000009085 0 1 1 00000300
2 1 -1 -1 000003106 0 1 1 00000320
86 1 -1 -1 0000033091 0 1 1 000003407 7 -1 -1 0000035089 0 1 1 00000360
00000370FORC 00000380
7 7 -102 000 0000039091 0 -102 000 00000300
00000480END 00000490
_MACR 00000500UTAN 000050FORM 00000510SOLV 00000520OISP 00000530STRE 0000050REAC 00000550REAC 00000560ETOP 00000570
-STOP 00000580
110
EL , ... . - - " - ... . .. -..
Input Dataset Run No. 3BRC4066 (FCtGqOUKO): OUTPUT FROM TSO XPRINT
AT 13:42:48 ON 12/05/80 - BRC4066.TESTZ.DATA
FEAP CROSS-CHANNEL FLOW--LINEAR NEWTONIAN//72 ELEMENTS 00000010253 72 1 2 2 8 0 00000020
COO.7 000000301 1 000 000 00000040
13 0 000 100 0000005014 1.08333300 000 0000006020 0.083333300 100 0000007021 1.166666700 000 0000008033 0.166666700 100 0000009034 1 .2500 000 0000010040 0 .25DO 100 0000011041 1.333333300 000 0000012053 0.333333300 100 0000013054 1.416666700 000 0000014060 0.4166667D0 100 0000015061 1 .500 OD 0000016073 0 .500 100 0000017074 1.583333300 000 0000018080 0.5333333C0 100 000C019081 1.666666700 000 0000010093 0.666666700 100 00000210
94 1 .7500 OD 00000220100 0 .7500 100 00000230101 1.833333300 000 00000240113 0.833333300 100 00000:50114 1.916666700 000 00000260120 0.916666700 ID0 00000270121 1 100 O0 00000280133 0 100 100 00000290134 11.08333300 000 00000300140 01.05333300 100 00000310141 11.16666700 000 00000320153 01.16666720 100 00000330154 1 1.2500 000 00000340160 0 1.2500 100 00000350161 11.333333'0 OD 00000360173 01.33333300 100 00000370174 11.41666/00 000 00000380180 01.41666700 100 00000390181 1 1.500 OD 00000400193 0 1.500 100 00000410194 11.58333300 OD 00000420200 01.583333!00 100 00000430201 11.66666700 000 00000440213 01.66666700 100 00000450214 1 1.7500 00 00000460220 0 1.7500 100 00000470221 11.83333300 00 00000480233 01.83333300 100 00000490234 11.91666700 000 00000500240 01.91666700 100 00000510241 1 200 000 00000520253 0 2D0 100 00000530
00000540ELEM 00000550
1 1 1 21 23 3 14 22 15 2 20 00000560
'!
111
13 1 3 23 2S 5 15 24 16 4 20 0000057025 1 5 25 27 7 16 26 17 6 20 0000058037 1 7 27 29 9 17 28 18 8 20 0000059049 1 9 29 31 11 16 30 19 10 20 0000060061 1 11 31 33 13 19 32 20 12 20 00000610
00000620MATE 00000630
1 5 EIGHT-NODE SERENDIPITY PENALTY ELEMENT 000006401 0 1 1 1.0 000006502 .1000.009 .7900+003 .0000 00000660
00000670so0" 00000680
1 1 -1 -1 0000069013 0 1 1 0000070014 20 -1 -1 00000710234 0 1 1 000007Z0
21 20 -1 -1 000007302zi1 0 1 1 00000740241 1 -1 -1 00000750253 0 1 1 0000076020 20 -1 -1 00C007702 0 0 1 1 0000078033 20 -1 -1 00000790233 0 1 1 00000800
00000810FCqC 00000820
20 i0 -102 000 00000830240 0 -102 000 0000084013 20 -102 00 00000850
253 0 -102 000 0000086000000870
E1D 00000880MACR 00C00090TANG 000009C0FC;H? 00000910SOLV 00000920ISP 00000930
STRE 00000940END 00000950STOP 00000960
11
112
?A
Input Dataset Run No. 4BPC4C66 (FCREG;OU, ): OUTPUT FROM TSO XPRItT
AT 16:22:13 ON 12/05/80 - 5RC4066.TEST.DATA
FEAP CROSS-CHANNEL FLOW - NEWJTONZAN WITH CONVECTION 0000001091 18 1 2 2 9 0 00000020
COR 000000301 7 000 oo 0000004065 0 too COO 000000502 7 000.166666700 00000060
86 0 200.166666700 000000703 7 000.333333330 0000008087 0 2D0.333333300 000000904 7 000 .500 000001008 0 200 .500 00000110S 7 000.666666700 0000012089 0 200.666666700 000001306 7 000.833333300 00000140
c0 0 200.833333300 000001507 7 ODO 100 00000160
91 0 t00 100 0000017000000180
ELEHt 00000190S 1 115 17 3 8 16 10 2 9 14 0 c0002007 1 3 17 19 5 10 18 12 4 11 14 00000210
13 1 5 19 21 7 12 20 14 6 13 14 0000022000000230
MATE 000002401 5 NINE-NODE LAGRAGIAN PENALTY ELEMENT 000002501 0 1 1 1.0 000002602 .1000+009 .7900+003 1.6000 00004270
00000275BOUN4 00000230
1 ? -1 -1 0000029065 0 1 1 000003002 1 -1 -1 000003106 0 1 1 0000032086 1 -1 -1 0000033091 0 1 1 000003407 7 -1 -1 00000350
89 0 1 1 0000036000000370
FONC 000003807 7 -102 oo 00000390
91 0 -102 000 0000040000000480
END) 00000490_MACR 000cOSO0OT 1. 00000505LOOP 3 00000520UTAH 00000530FORM 00000540SOLV 00000550OISP 1 00000560STRE 1 00000565TIME 00000567
.:NtXT 00000570DISP 00000580STRE 00000590REAC 00000595
113
A 2- .
ENDSTOP 00000600
00000610
114
V
Input Dataset Run No. 6BRC4066 (FCOEGROUN ): OUTPUT FROM 750 XPRINT
AT 13:50:54 ON 12/07/80 - EPC4066.TEST.OATA
FEAP SQUARE CAVITY- OLOROYD VISCOELASTIC (RHO:O, P4z1. WSZ.011 00000010
91 18 1 2 2 9 0 00000020
COON 00000030
1 7 o0o ODO 00000040
85 0 200 000 000000SO
2 7 000.166666700 00000060
86 0 200.166666790 00000070
3 7 000.3333333DO 00000080
87 0 200.3333333D0 00000090
4 7 000 .SOO 00000100
88 0 200 .500 00000110
S 7 000.66666670 00000120
89 0 200.666666700 00000130
6 7 0D0.833333300 00000140
90 0 200.833333300 000001SO
7 7 000 100 00000160
91 0 200 100 0000017000000180
ELEM 00000190
1 1 1 15 17 3 8 16 10 2 9 14 00000200
7 1 3 17 19 5 10 18 12 4 11 14 00000210
13 1 5 19 21 7 12 20 14 6 13 14 000022000000230
MATE 00000240
1 5 NINE-NODE LAGRAGIAN PEHALTY ELEMENT 00000250
1 0 3 1 1.0 00000260
2 .1000.009 .79004003 .3950+007 0000027000000275
BOUN 00000280
1 7 -1 -1 00000290
85 0 1 1 00000300
2 1 -1 -1 00000310
6 0 1 1 00000320
86 1 -1 -1 00000330
91 0 1 1 00000330
7 7 -1 -1 00000350
89 0 1 1 0000036000000370
FORC 00000380
7 7 -102 000 00000390
91 0 -102 000 0000040000000485
END 00000490
EANR 00000500" OT 1. 00000510
LOOP 20 00000520
UTAN 00000530
FORH 00000540
SOLV 00000550
OISP 5 00000558
STRE 5 00000566T!ME 00000575
,TIME 000005Sa
.NEXT 00000585
DISP 00000588
STRE 00000s91
REAC 0009
115
.E~c ooooosg
i15 -
- ..-
END 00000600STOP 00000610
116
Input Dataset Run No. 13ORC4066 PFOEORqOJNO): OUTPUT FROM TSO XPRINT
AT 15%38:57 ON 12/07/00 - ERC4066.TEST3.0ATA
FEAP E4TRY FLOW - OLOROYD VXSCOEILASTZC (RHOa@, P41, 1SzO.0005) 00000010121 94 1 2 2 9 0 60000024
COOR 000006301 1 000 000 000000409 0 000 100 00000050
10 1 .12500 000 0000060010 0 .12500 100 0000007019 1 .2so0 *DO 000000627 0 .2SOO 100 0000009020 1 .37500 000 0000410036 0 .37500 100 0000021637 1 .0 000 0000012045 0 .500 100 0000013046 1 .62500 000 000001*054 0 .6250 100 000001$055 1 .7500 100 0000016063 0 .7500 100 0000017064 1 .87500 oo 0000018072 0 .87500 10 0000019073 1 100 000 00000:0081 0 100 100 0000021082 5 1.12500 .2500 00000220117 0 200 .2500 0000023083 5 1.12500 .37S0 00000240
118 0 zo .37500 0000025084 5 1.12500 .500- 00000260119 0 200 .500 0000027085 5 2.1230 .62500 00000260120 0 200 .62500 0000029086 S 1.12500 .7500 00000300121 0 DO .7500 00000310
00000320ELEM 00000330
1 1 1 19 i 3 10 20 12 2 11 18 000003*0S 1 3 21 23 S 12 22 14 4 13 18 000003509 1 5 23 25 7 14 t4 16 6 1s 10 0000036013 1 7 25 27 9 16 26 10 0 17 18 0000037017 1 75S 8709 77 82 6 4 76 83 0 0000038018 1 87 97 99 69 92 90 94 80 93 10 0000039021 1 77 89 91 79 84 90 86 78 S 0 0000040022 1 89 99 101 91 94 100 96 90 95 10 00000410
00000420HATE 00000430
1 5 NINE-NODE LAGRANGE PENALTY ELEMENT 000004401 0 3 1 1.0 00000*S02 .1000#009 .79004003 .0000 .7950+008 00000460
00000470BM 00000480
1 9 -1 -1 00000490
73 0 1 1 000005002 1 -1 -1 00000S10a 0 1 1 00000S20
- 9 9 -1 -1 0000053081 0 1 1 00000S4074 0 1 1 0000055075 0 1 1 OO0006
117
A, -.
79 0 1 1 00000570
80 0 1 1 00000580
82 5 -1 -1 00000590
117 0 1 1 00000600
86 S -1 -1 00000610
121 0 1 1 00000620
118 0 0 1 00000630
119 0 0 1 00000640
120 0 0 1 0000066000000670
FC C 000006802 1 102 00 00000670
8 0 102 000 00000690000080
EIO 00000710ENCR 00000720
OT 1. 00000730
LOOP 30 000007S0
UTAH 00000750F0M 00000760
SOLV 00000770
OrSP 5 00000780
STPE 5 00000790
TIME 00C00800NEXT 00000810
OISP 00000820
STRE 00000830
REAC 00000840REND 00000850EISID 00000860STOP
liI
118
777~-
8RC4066 tFOREGROUN0): OUTPUT FROM TSO XFRINT Input Dataset Run No. 20AT 18:59:50 ON 12/18/00 - BRC4066.TEST3.OATA
FEAP ENTRY FLOW - OLDROYD VIVOELASTIC (RHO=O, P4=1, &ISzO.0005) 00000010121 24 1 2 2 9 0 00000020COOR
000000301 1 000 000 000000409 0 000 100 0000003010 2 .12500 oo 0000006018 0 .12500 100 00000070
19 1 .2S04 000 0000008027 0 .2500 100 0000009028 1 .37500 000 0000010036 0 .37500 100 0000031037 1 .sDo ODO 0000012045 0 .500 100 0000013046 1 .62500 000 0000014054 0 .62500 100 0000015035 1 .7500 000 0000016063 0 .7SO 100 0000017064 1 .87500 000 0000018072 0 .87500 100 0000019073 1 100 000 0000020081 0 100 100 0000021082 5 1.12500 .2500 00000220117 0 Zo .2500 0000023083 S 1.12500 .37500 00000240118 0 200 .37500 0000025084 5 1.12500 .500 00000260119 0 ZOO .500 0000027085 5 1.12500 .62500 00000280120 0 v00 .62500 0000029086 5 1.12500 .7500 00000300
121 0 200 .7500 00000310
ELE"f 00000320000003301 1 1 19 21 3 10 20 12 2 11 18 000003405 1 3 21 23 S 12 22 14 4 13 18 000003309 1 5 23 25 7 14 24 16 6 15 18 0000036013 1 7 25 27 9 16 26 18 8 17 18 0000037017 1 75 87 89 77 82 88 84 76 83 0 0000038018 1 87 97 99 89 92 98 94 88 93 10 0000039021 1 77 89 91 7984 90 86 7885 0 0000040022 1 89 99 101 91 94 100 96 90 95 10 00000410
MHATE 00000420000004301 5 NINE-NODE LAGRANGE PENALTY ELEMENT 000004401 0 1 1 1.0 000004502 .1000+009 .79004003 .0000 .7950+008 00000460
BOtH 000004701 9 -000004801 9 -1 -1 0000049073 0 2 1 0000050075 0 1 1 00000503
- 79 0 1 1 00000306- 2 1 0 -1 000005102 0 0 1 000005209 9 -1 -1 00000530
81 0 1 1 00000540
119
74 0 1 1 0000055070 0 1 1 0000050080 0 1 1 000059082 5 -1 -1 00000590117 0 1 1 0000061086 S -1 -1 00000620
121 0 1 1 00000630118 0 0 1 00000640119 0 0 00000650120 0 0 1 00000660
000006702 a 4.202 000 00000680
2.102 000006814 0 4.202 00000682
5 0 2.102 000006840 4.202 00000685
7 0 2.102 00000690S 0 4.202 000 00000700
00000710END 00000720MACR 00000750UTAH 000C0760FOR" 00000770SOLV 00000780OisP 00000790STRE 00000840REAC 00000850END 00000060STOP
120
APPENDIX 5
Brief Review of Gyroscope Theory
This Appendix is presented for the benefit ot the materials
engineer who may not be familiar with the theory of gyroscopic
behavior. The discussion is taken entirely from Wrigley et. al.
[36]. Figure 18 shows a cutaway of the single degree of freedom
gyroscope used in this study. The normal assumptions for the
description of a gyro element performance are:
1. The rotor spins about an axis of symmetry.
2. The rotor spins at constant speed.
3. Spin angular momentum is much greater than non-spin
angular momentum.
4. Center of mass of the rotor and gyro element coincide,
and 5. The rotor bearing structure is rigid.
For a platform stabilized single degree of freedom gyro,
these assumptions lead to the performance equation:
I A" +c + k H + U(MOA)
dt2 +gdt g S1 IA (cmd SRA H ]Igg ME-dwoA
For integrating gyros, a restraining torsional spring is
eliminated, hence k = 0 and the performance equation becomes:g
Td 2 e d!e .HF 4W-Ut( MOA)] - wOg +H7 d g IA cmd O SRA H -i g t
or d e) g+ - + Hs dt)
121
-5
Therefore:
g HE + 9 IA cmd - SRA H / g - O
Where
e - Output axis rotation
WIA -Input axis angular rate
W Commanded output axis angular rate
S Spin reference axis angular rate
Hs Rotor angular momentum
T 9 I/c E time constant
I -gyro output axis effective moment of inertia
cg float damping coefficient
U(MOA) Uncertain torque aboat output axis
Assuming & T << &WIA cmd, the eguation becomes
T !L' + ( = g U MOA)
This equation shows that the gyro drift uncertainty is a first
order response to the time integral of the uncertainty torques
about the output axis.
Alternately expressing the equation in terms of drift rate:
dwOA U(14 OA)Tg -- t-+ wOA - c--9
Hence, any source of uncertain torque of the torque sunning
member about the output axis is a contributor to the possible
inaccuracy of the gyro element.
122
MELi
The most common sources of these uncertainties are
gimbal friction and mass unbalance.
These are factors very sensitive to the material state
and processing variables. It is for this reaso. that a
rational method of selecting injection molding parameters is
required. A brief example of this is presented.
Prior to introduction into service, the molded gyro is
balanced. Remaining unbalance can be nullified by compensa-
tion in the feedback loop of the control system. However,
from the drift rate equation we can see that for a step
acceleration the steady state (t ®) drift rate, due to
torque uncertainties caused by variations to the balance, is:
WOAI VeagT
s.S. Cg
Where p is the mass density, V is the effective volume
of unbalance, e is the amount of mass eccentricity, and a is
the step acceleration in g's.
Taking typical values:
C 20588 dyne-sec
T= 0.0017 secg
WOAI = 1/hr = 4.8 x 10-6 rad/sec8.3.
p = 1.6 gm/cm (Polypheneline Sulfide)
123
I .. .._. .
We obtain:
a cm
For an acceleration of lOgs then we get:
Ve - 0.00363 cm4
which defines the bounds of mass unbalance which can be tolerated,
for the specified performance, due to long term materials be-
havior (creep relaxation, non-uniform thermal strain, etc.).
124
O .
BIBLIOGRAPHY
1. Zienkiewicz, O.C., The Finite Element Method, Chap. 22,
3rd ed., McGraw-Hill, 1977.
2. Chung, T.J., Finite Element Analysis in Fluid Dynamics,
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3. Bathe, K.J., Finite Element Procedures in Engineering
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6. Finlayson, B.A., "Weighted Residual Methods and Their
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126
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127
21. Malvern, L.E., Introduction to the Mechanics of a
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128
31. Lax, P.D., and Richtmyer, R.D., "Survey of the Stability
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129